sign charts of factorable formulas

1. Sign Charts of Factorable Formulas
Math 260
Dr. Frank Ma
LA Harbor College

2. Given an expression f, it’s important to identify when
the output is positive (f > 0) and when the output is
negative (f < 0) when f is evaluated with a value x.
Sign Charts of Factorable Formulas

3. Example A. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2, –1/2.
Given an expression f, it’s important to identify when
the output is positive (f > 0) and when the output is
negative (f < 0) when f is evaluated with a value x.
Sign Charts of Factorable Formulas

4. For polynomials or rational expressions,
factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when
the output is positive (f > 0) and when the output is
negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2, –1/2.
Sign Charts of Factorable Formulas

5. In factored form x2 – 2x – 3 = (x – 3)(x + 1)
For polynomials or rational expressions,
factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when
the output is positive (f > 0) and when the output is
negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2, –1/2.
Sign Charts of Factorable Formulas

6. In factored form x2 – 2x – 3 = (x – 3)(x + 1)
Hence, for x = –3/2:
(–3/2 – 3)(–3/2 + 1)
For polynomials or rational expressions,
factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when
the output is positive (f > 0) and when the output is
negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2, –1/2.
Sign Charts of Factorable Formulas

7. In factored form x2 – 2x – 3 = (x – 3)(x + 1)
Hence, for x = –3/2:
(–3/2 – 3)(–3/2 + 1) is (–)(–) = + .
For polynomials or rational expressions,
factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when
the output is positive (f > 0) and when the output is
negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2, –1/2.
Sign Charts of Factorable Formulas

8. In factored form x2 – 2x – 3 = (x – 3)(x + 1)
Hence, for x = –3/2:
(–3/2 – 3)(–3/2 + 1) is (–)(–) = + .
And for x = –1/2:
(–1/2 – 3)(–1/2 + 1)
For polynomials or rational expressions,
factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when
the output is positive (f > 0) and when the output is
negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2, –1/2.
Sign Charts of Factorable Formulas

9. In factored form x2 – 2x – 3 = (x – 3)(x + 1)
Hence, for x = –3/2:
(–3/2 – 3)(–3/2 + 1) is (–)(–) = + .
And for x = –1/2:
(–1/2 – 3)(–1/2 + 1) is (–)(+) = – .
For polynomials or rational expressions,
factor them to determine the signs of their outputs.
Given an expression f, it’s important to identify when
the output is positive (f > 0) and when the output is
negative (f < 0) when f is evaluated with a value x.
Example A. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2, –1/2.
Sign Charts of Factorable Formulas

10. Example B. Determine whether the outcome is
x2 – 2x – 3
x2 + x – 2
x2 – 2x – 3
x2 + x – 2
In factored form =
(x – 3)(x + 1)
(x – 1)(x + 2)
if x = –3/2, –1/2.
+ or – for
Sign Charts of Factorable Formulas

11. x2 – 2x – 3
x2 + x – 2
In factored form =
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x = –3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(–)
(–)(+)
Example B. Determine whether the outcome is
x2 – 2x – 3
x2 + x – 2
if x = –3/2, –1/2.
+ or – for
Sign Charts of Factorable Formulas

12. x2 – 2x – 3
x2 + x – 2
In factored form =
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x = –3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(–)
(–)(+)
< 0
Example B. Determine whether the outcome is
x2 – 2x – 3
x2 + x – 2
if x = –3/2, –1/2.
+ or – for
Sign Charts of Factorable Formulas

13. x2 – 2x – 3
x2 + x – 2
In factored form =
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x = –3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(–)
(–)(+)
< 0
For x = –1/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(+)
(–)(+)
Example B. Determine whether the outcome is
x2 – 2x – 3
x2 + x – 2
if x = –3/2, –1/2.
+ or – for
Sign Charts of Factorable Formulas

14. x2 – 2x – 3
x2 + x – 2
In factored form =
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x = –3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(–)
(–)(+)
< 0
For x = –1/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(+)
(–)(+)
> 0
Example B. Determine whether the outcome is
x2 – 2x – 3
x2 + x – 2
if x = –3/2, –1/2.
+ or – for
Sign Charts of Factorable Formulas

15. x2 – 2x – 3
x2 + x – 2
In factored form =
(x – 3)(x + 1)
(x – 1)(x + 2)
Hence, for x = –3/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(–)
(–)(+)
< 0
For x = –1/2:
(x – 3)(x + 1)
(x – 1)(x + 2)
=
(–)(+)
(–)(+)
> 0
This leads to the sign charts of formulas. The sign–
chart of a formula gives the signs of the outputs.
Example B. Determine whether the outcome is
x2 – 2x – 3
x2 + x – 2
if x = –3/2, –1/2.
+ or – for
Sign Charts of Factorable Formulas

16. Here is an example, the sign chart of f = x – 1:
Sign Charts of Factorable Formulas

17. Here is an example, the sign chart of f = x – 1:
1
f = 0 + +
– – – – x – 1
Sign Charts of Factorable Formulas

18. Here is an example, the sign chart of f = x – 1:
1
f = 0 + +
– – – – x – 1
The "+" indicates the region where the output is
positive i.e. if 1 < x.
Sign Charts of Factorable Formulas

19. Here is an example, the sign chart of f = x – 1:
1
f = 0 + +
– – – – x – 1
The "+" indicates the region where the output is
positive i.e. if 1 < x. Likewise, the "–" indicates the
region where the output is negative, i.e. x < 1.
Sign Charts of Factorable Formulas

20. Construction of the sign–chart by Sampling
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +
– – – – x – 1
The "+" indicates the region where the output is
positive i.e. if 1 < x. Likewise, the "–" indicates the
region where the output is negative, i.e. x < 1.
Sign Charts of Factorable Formulas

21. Construction of the sign–chart by Sampling
I. Solve for f = 0 (and denominator = 0) if there is any
denominator.
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +
– – – – x – 1
The "+" indicates the region where the output is
positive i.e. if 1 < x. Likewise, the "–" indicates the
region where the output is negative, i.e. x < 1.
Sign Charts of Factorable Formulas

22. Construction of the sign–chart by Sampling
I. Solve for f = 0 (and denominator = 0) if there is any
denominator.
II. Draw the real line, mark off the answers from I.
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +
– – – – x – 1
The "+" indicates the region where the output is
positive i.e. if 1 < x. Likewise, the "–" indicates the
region where the output is negative, i.e. x < 1.
Sign Charts of Factorable Formulas

23. Construction of the sign–chart by Sampling
I. Solve for f = 0 (and denominator = 0) if there is any
denominator.
II. Draw the real line, mark off the answers from I.
III. Sample each segment for signs by testing a point
in each segment.
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +
– – – – x – 1
The "+" indicates the region where the output is
positive i.e. if 1 < x. Likewise, the "–" indicates the
region where the output is negative, i.e. x < 1.
Sign Charts of Factorable Formulas

24. Construction of the sign–chart by Sampling
I. Solve for f = 0 (and denominator = 0) if there is any
denominator.
II. Draw the real line, mark off the answers from I.
III. Sample each segment for signs by testing a point
in each segment.
Here is an example, the sign chart of f = x – 1:
1
f = 0 + +
– – – – x – 1
The "+" indicates the region where the output is
positive i.e. if 1 < x. Likewise, the "–" indicates the
region where the output is negative, i.e. x < 1.
Fact: The sign stays the same for x's in between the
values from step I (where f = 0 or f is undefined.)
Sign Charts of Factorable Formulas

25. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Sign Charts of Factorable Formulas

26. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Sign Charts of Factorable Formulas

27. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1)
4
–1
Sign Charts of Factorable Formulas

28. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1)
Select points to sample in each segment:
4
–1
Sign Charts of Factorable Formulas

29. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1)
4
–1
Select points to sample in each segment:
Test x = – 2,
–2
Sign Charts of Factorable Formulas

30. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1)
4
–1
Select points to sample in each segment:
Test x = – 2,
get – * – = + .
Hence the segment
is positive. Draw +
sign over it.
–2
Sign Charts of Factorable Formulas

31. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1) + + + + +
4
–1
Select points to sample in each segment:
Test x = – 2,
get – * – = + .
Hence the segment
is positive. Draw +
sign over it.
–2
Sign Charts of Factorable Formulas

32. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1) + + + + +
0 4
–1
Select points to sample in each segment:
Test x = – 2,
get – * – = + .
Hence the segment
is positive. Draw +
sign over it.
–2
Test x = 0,
get – * + = –.
Hence this segment
is negative.
Put – over it.
Sign Charts of Factorable Formulas

33. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1) + + + + + – – – – –
0 4
–1
Select points to sample in each segment:
Test x = – 2,
get – * – = + .
Hence the segment
is positive. Draw +
sign over it.
–2
Test x = 0,
get – * + = –.
Hence this segment
is negative.
Put – over it.
Sign Charts of Factorable Formulas

34. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1) + + + + + – – – – –
0 4
–1
Select points to sample in each segment:
Test x = – 2,
get – * – = + .
Hence the segment
is positive. Draw +
sign over it.
–2
Test x = 0,
get – * + = –.
Hence this segment
is negative.
Put – over it.
Test x = 5,
get + * + = +.
Hence this segment
is positive.
Put + over it.
5
Sign Charts of Factorable Formulas

35. Example C. Let f = x2 – 3x – 4 , use the sign–
chart to indicate when is f = 0, f > 0 and f < 0.
Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0  x = 4 , –1
Mark off these points on a line:
(x–4)(x+1) + + + + + – – – – – + + + + +
0 4
–1
Select points to sample in each segment:
Test x = – 2,
get – * – = + .
Hence the segment
is positive. Draw +
sign over it.
–2
Test x = 0,
get – * + = –.
Hence this segment
is negative.
Put – over it.
Test x = 5,
get + * + = +.
Hence this segment
is positive.
Put + over it.
5
Sign Charts of Factorable Formulas

36. Example D. Make the sign chart of f =
(x – 3)
(x – 1)(x + 2)
Sign Charts of Factorable Formulas

37. Example D. Make the sign chart of f =
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3.
Sign Charts of Factorable Formulas

38. Example D. Make the sign chart of f =
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3. The zeroes of the denominator
x = 1, –2 are the values where f is undefined (UDF).
Sign Charts of Factorable Formulas

39. Example D. Make the sign chart of f =
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3. The zeroes of the denominator
x = 1, –2 are the values where f is undefined (UDF).
Mark these values on a real line.
Sign Charts of Factorable Formulas

40. Example D. Make the sign chart of f =
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3. The zeroes of the denominator
x = 1, –2 are the values where f is undefined (UDF).
Mark these values on a real line.
(x – 3)
(x – 1)(x + 2) –2 1 3
UDF UDF f=0
Sign Charts of Factorable Formulas

41. Example D. Make the sign chart of f =
Select a point to sample in each segment:
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3. The zeroes of the denominator
x = 1, –2 are the values where f is undefined (UDF).
Mark these values on a real line.
(x – 3)
(x – 1)(x + 2) –2 1 3
UDF UDF f=0
–3 0 2 4
Sign Charts of Factorable Formulas

42. Example D. Make the sign chart of f =
Select a point to sample in each segment:
Test x = –3,
we've a
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3. The zeroes of the denominator
x = 1, –2 are the values where f is undefined (UDF).
Mark these values on a real line.
(x – 3)
(x – 1)(x + 2) –2 1 3
UDF UDF f=0
–3
( – )
( – )( – )
= –
segment.
0 2 4
Sign Charts of Factorable Formulas

43. Example D. Make the sign chart of f =
Select a point to sample in each segment:
Test x = –3,
we've a
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3. The zeroes of the denominator
x = 1, –2 are the values where f is undefined (UDF).
Mark these values on a real line.
(x – 3)
(x – 1)(x + 2) –2 1 3
UDF UDF f=0
–3
( – )
( – )( – )
= –
segment.
0 2 4
Test x = 0,
we've a
( – )
( – )( + )
= +
segment.
Sign Charts of Factorable Formulas

44. Example D. Make the sign chart of f =
Select a point to sample in each segment:
Test x = –3,
we've a
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3. The zeroes of the denominator
x = 1, –2 are the values where f is undefined (UDF).
Mark these values on a real line.
(x – 3)
(x – 1)(x + 2) –2 1 3
UDF UDF f=0
–3
( – )
( – )( – )
= –
segment.
0 2 4
Test x = 0,
we've a
( – )
( – )( + )
= +
segment.
Test x = 2,
we've a
( – )
( + )( + )
segment.
= –
Sign Charts of Factorable Formulas

45. Example D. Make the sign chart of f =
Select a point to sample in each segment:
Test x = –3,
we've a
(x – 3)
(x – 1)(x + 2)
The root for f = 0 is from the zero of the numerator
which is x = 3. The zeroes of the denominator
x = 1, –2 are the values where f is undefined (UDF).
Mark these values on a real line.
(x – 3)
(x – 1)(x + 2) –2 1 3
UDF UDF f=0
–3
( – )( – )
= –
segment.
0 2 4
Test x = 0,
we've a
( – )
( – )( + )
= +
segment.
Test x = 2,
we've a
( – )
( + )( + )
segment.
= –
Test x = 4,
we've a
( + )
( + )( + )
segment.
= +
– – – – + + + – – – + + + +
( – )
Sign Charts of Factorable Formulas

46. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
Sign Charts of Factorable Formulas

47. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
Sign Charts of Factorable Formulas

48. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16)
Sign Charts of Factorable Formulas

49. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
Sign Charts of Factorable Formulas

50. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
Sign Charts of Factorable Formulas

51. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable
Sign Charts of Factorable Formulas

52. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable
Sign Charts of Factorable Formulas

53. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.
Sign Charts of Factorable Formulas

54. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
The order of a root is the corresponding power
raised in the factored form, i.e.
the order of the r1 is N1, order of r2 is N2 ,etc..
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.
Sign Charts of Factorable Formulas

55. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
The order of a root is the corresponding power
raised in the factored form, i.e.
the order of the r1 is N1, order of r2 is N2 ,etc..
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.
x = 0 has order 3,
Sign Charts of Factorable Formulas

56. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
The order of a root is the corresponding power
raised in the factored form, i.e.
the order of the r1 is N1, order of r2 is N2 ,etc..
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.
x = 0 has order 3, x = –2 and x = 2 have order 2.
Sign Charts of Factorable Formulas

57. In this section we give a theorem about sign-charts
of factorable polynomials and rational expressions.
A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
The order of a root is the corresponding power
raised in the factored form, i.e.
the order of the r1 is N1, order of r2 is N2 ,etc..
N1 N2 Nk
Example E. P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.
x = 0 has order 3, x = –2 and x = 2 have order 2.
Sign Charts of Factorable Formulas

58. An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
Sign Charts of Factorable Formulas

59. An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
Sign Charts of Factorable Formulas

60. An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)
Sign Charts of Factorable Formulas

61. An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)
b. x = 2 or –2 are even-ordered roots
(each has order 2).
Sign Charts of Factorable Formulas

62. An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)
b. x = 2 or –2 are even-ordered roots
(each has order 2).
Theorem (The Even/Odd–Order Sign Rule)
Sign Charts of Factorable Formulas

63. An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)
b. x = 2 or –2 are even-ordered roots
(each has order 2).
Theorem (The Even/Odd–Order Sign Rule)
For the sign-chart of a factorable polynomial
1. the signs are the same on both sides of
an even-ordered root,
2. the signs are different on two sides of
an odd-ordered root.
Sign Charts of Factorable Formulas

64. An important property of a root is whether it is an
even-ordered root or an odd-ordered root.
For example, for P(x) = 2x3(x + 2)2(x – 2)2,
a. x = 0 is an odd-ordered root (its order is 3)
b. x = 2 or –2 are even-ordered roots
(each has order 2).
Theorem (The Even/Odd–Order Sign Rule)
For the sign-chart of a factorable polynomial
1. the signs are the same on both sides of
an even-ordered root,
2. the signs are different on two sides of
an odd-ordered root.
This theorem simplifies the construction of sign-charts
and graphs (later) of factorable polynomials.
Sign Charts of Factorable Formulas

65. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
Sign Charts of Factorable Formulas

66. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3.
Sign Charts of Factorable Formulas

67. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Sign Charts of Factorable Formulas

68. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)
x=-3 (even)
Sign Charts of Factorable Formulas

69. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)
x=-3 (even)
Sample a point, say x = 4 and we get P(4) positive.
Sign Charts of Factorable Formulas

70. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)
x=-3 (even)
Sample a point, say x = 4 and we get P(4) positive.
Mark the segment as such.
+
x=4
Sign Charts of Factorable Formulas

71. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)
x=-3 (even)
Sample a point, say x = 4 and we get P(4) positive.
Mark the segment as such. By the theorem, across
the root x = 3 the sign changes to "–" because x = 3
is odd-ordered.
+
x=4
Sign Charts of Factorable Formulas

72. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)
x=-3 (even)
Sample a point, say x = 4 and we get P(4) positive.
Mark the segment as such. By the theorem, across
the root x = 3 the sign changes to "–" because x = 3
is odd-ordered.
change
sign +
x=4
Sign Charts of Factorable Formulas

73. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)
x=-3 (even)
Sample a point, say x = 4 and we get P(4) positive.
Mark the segment as such. By the theorem, across
the root x = 3 the sign changes to "–" because x = 3
is odd-ordered. Similarly, across the root x = 0, the
sign changes again to "+".
change
sign
change
sign
+ +
x=4
Sign Charts of Factorable Formulas

74. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)
x=-3 (even)
Sample a point, say x = 4 and we get P(4) positive.
Mark the segment as such. By the theorem, across
the root x = 3 the sign changes to "–" because x = 3
is odd-ordered. Similarly, across the root x = 0, the
sign changes again to "+". But across x = -3 the
sign stays as "+" because it is even-ordered
change
sign
change
sign
+ +
x=4
Sign Charts of Factorable Formulas

75. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)
x=-3 (even)
Sample a point, say x = 4 and we get P(4) positive.
Mark the segment as such. By the theorem, across
the root x = 3 the sign changes to "–" because x = 3
is odd-ordered. Similarly, across the root x = 0, the
sign changes again to "+". But across x = -3 the
sign stays as "+" because it is even-ordered
change
sign
change
sign
+
sign
unchanged
+ +
x=4
Sign Charts of Factorable Formulas

76. Example F. Make the sign-chart of x3(x + 3)2 (x – 3)
The roots are x = 0, –3 and 3. x = 0 and 3 are odd-
ordered roots and root x = –3 is an even-ordered root.
Draw a line, mark off these roots and their types.
x=0 (odd) x=3 (odd)
x=-3 (even)
Sample a point, say x = 4 and we get P(4) positive.
Mark the segment as such. By the theorem, across
the root x = 3 the sign changes to "–" because x = 3
is odd-ordered. Similarly, across the root x = 0, the
sign changes again to "+". But across x = -3 the
sign stays as "+" because it is even-ordered
and the chart is completed.
change
sign
change
sign
+
sign
unchanged
+ +
x=4
Sign Charts of Factorable Formulas

77. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Sign Charts of Factorable Formulas

78. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Sign Charts of Factorable Formulas

79. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

80. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

81. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

82. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered).
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

83. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered).
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

84. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ".
x=0 (even) x=2 (odd)
x= 1 (even) x=3
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

85. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)
x= 1 (even) x=3
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

86. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)
x= 1 (even)
+ change
x=3
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

87. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)
x= 1 (even)
+ change
unchanged
+
x=3
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

88. The theorem may be generalized to rational formulas
that are factorable, that is, both the numerator and
the denominator are factorable.
Factor the expression: 2x2 – x3
x2 – 2x + 1
= x2(2 – x)
(x – 1)2
Roots of the numerator are x = 0 (even-ordered)
and x = 2 (odd-ordered). The root of the
denominator is x = 1 (even-ordered). Draw and
test x = 3, we get " – ". Complete the sign-chart.
x=0 (even) x=2 (odd)
x= 1 (even)
+ change
unchanged
unchanged
+
+
x=3
Example G. Draw the sign–chart of 2x2 – x3
x2 – 2x + 1
Sign Charts of Factorable Formulas

89. 1. x2(x – 2) 2. x2(x – 2)2 3. x(x – 2)2
4. x3(x – 2) 5. x(x + 2)2(5 – x) 6. x2(x + 2)2(5 – x)
8. 9x2 – x4
7. x2(x + 2)2(x – 5)3 9. x4 – 4x3 + 4x2
11. 3x2(2 – x)7(x – 1)4
10. 3(2x– 5) 2(x + 2)7(x – 1)4
12. (5 – x )2(3 – x)7(2x – 1)5
B. Draw the sign–chart of each formula below.
1.
7.
2. 3.
4. 5. 6.
x2 – 4
x2 – 4x + 4
x – 4
x2
(x + 3)2
x + 4
x + 2
x2 – 3x + 2
x2(x – 2)3
(x + 4)2(x + 2) 8.
x(x – 4)3
(x2 + 4)(x + 4)2
Exercise A. Draw the sign–chart of each formula below.
x + 4
x2 x2(x – 4)3
x2(2 + x)2
Sign Charts of Factorable Formulas

90. (Answers to odd problems) Exercise A.
1. 𝑥2
(𝑥 – 2)
x=0 x=2
+
3. 𝑥(𝑥 − 2)2
x=0 x=2
+
+
5. 𝑥(𝑥 + 2)2(5 – 𝑥)
x= – 2 x=5
x= 0
+
Sign Charts of Factorable Formulas

91. 11. 3𝑥2 2 − 𝑥 7(𝑥 − 1)4
x= 0 x=2
x= 1
+
7. (𝑥 + 2)2(𝑥 − 5)3
x= – 2 x=5
x= 0
+
+
+
9. 𝑥2 𝑥 − 2 2
x= 0 x=2
+
+
+
Sign Charts of Factorable Formulas

92. Exercise B.
1.
𝑥−4
𝑥+2
x= – 2 x=4
+
+
3.
𝑥2−4
𝑥+4
x= – 4 x=2
x= -2
+
+
UDF f=0
UDF f=0 f=0
5.
𝑥2(2+𝑥)2
𝑥2−3𝑥+2
+
UDF
x= –2 x=1
x= 0
+
x=2
UDF
f=0 f=0
+ +
7.
𝑥2(2+𝑥)2
𝑥2−3𝑥+2
+ UDF
x= 2
x= -2 x=0
x=-4
UDF f=0
f=0
+
+
Sign Charts of Factorable Formulas