Algebraic expressions, Monomials, coefficient, Like monomials, unlike monomials, Adding and subtracting monomials, Multiplying and dividing monomials, polynomial, degree, Adding and subtracting polynomial, Multiplying polynomial

1. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Algebraic expressions
Matematicas 2o E.S.O.
Alberto Pardo Milanes
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2. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
1 Monomials
2 Operations with monomials
3 Polinomials
4 Operations with polynomials
5 Multiplying polynomials
6 Exercises
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3. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Monomials
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4. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Monomials
Numerical value of an algebraic expression
An algebraic expression in variables x; y; z; a; r; t : : : k is an
expression constructed with the variables and numbers using
addition, multiplication, and powers.
To evaluate the numerical value of an algebraic expression means
that you have to replace the variable in the expression with values
and simplify the expression.
Example: To

5. nd the value of the algebraic expression x23x+4
if x = 3, you replace every x by 3 and simplify:
32 3 3 + 4 = 9 9 + 4 = 4.
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6. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Monomials
A number multiplied with a variable in an algebraic expression is
named coecient. The product of a coecient and one or more
variables is called a monomial.
Examples: x, 3xy2 and
2
5
x2y3z are all monomials, the coecients
are 1, 3 and
2
5
.
In a monomial with only one variable, the power is called its order,
or sometimes its degree. In a monomial with several variables, the
order/degree is the sum of the powers.
Examples: Deg(2x4)=4, Deg(7x3y2)=5.
Like monomials are monomials that have the exact same variables,
but dierent coecients. Unlike monomials are monomials that are
not like monomials.
Examples: 2x3y2 and
2
5
x3y2 are like monomials. 4xy2 and 4y2x4
are unlike monomials.
Alberto Pardo Milanes Algebraic expressions

7. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Operations with monomials
Alberto Pardo Milanes Algebraic expressions

8. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Operations with monomials
(Adding and subtracting monomials)
You can ONLY add and subtract like monomials. To add or
subtract like monomials, add or subtract the coecients and keep
the variables.
Examples: 3x+4x = (3+4)x = 7x, and 20a24a = (2024)a =
= 4a.
(Multiplying monomials)
To multiply monomials, multiply the coecients and add the
exponents with the same bases.
Examples: 3x2 5y = (3 5)x2 y = 15x2y, and 2a2 7ab4 =
= (2 7)a2 ab4 = 14a2+1b4 = 14a3b4.
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9. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Operations with monomials
(Dividing monomials)
To divide monomials, divide the coecients and subtract the
exponents with the same bases.
Example: 15x3y3z2 : 3xy3z = (15 : 3)x31y33z21 = 5x2z.
Alberto Pardo Milanes Algebraic expressions

10. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Polinomials
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11. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Polinomials
A polynomial is a mathematical expression involving a sum of
powers in one or more variables multiplied by coecients. A
polynomial in one variable is given by a sum of several monomials.
Example: 3x2 5x 2.
In a polynomial with only one variable, the highest power is called
its order, or sometimes its degree.
Example: Deg(x2 + 3x4 2x3 + 1) = Deg(3x4) = 4.
In a polynomial with several variables, the order/degree is the
highest sum of the powers of every term.
Example: Deg(xz3 + 3yx2z2 2) = Deg(yx2z2) = 5.
Alberto Pardo Milanes Algebraic expressions

12. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Operations with polynomials
Alberto Pardo Milanes Algebraic expressions

13. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Operations with polynomials
Add and subtract
(Adding polynomials)
Adding polynomials is just a matter of combining like monomials.
Remenber you can only add like monomials.
Example: (7x2x+4)+(x22x+3) = 7x2x+4+x22x+3 =
= 9x2 3x + 7.
(Subtracting polynomials)
To subtract a polynomial use the opposite of every coecient of
the subtrahend and add like monomials.
Example: (x3 + 3x2 + 5x 4) (3x3 8x2 5x + 6) =
= x3 + 3x2 + 5x 4 3x3 + 8x2 + 5x 6 =
= 2x3 + 11x2 + 10x 10
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14. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Multiplying polynomials
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15. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Multiplying polynomials
(Multiply polynomials)
To multiply two polynomials, we multiply each monomial of one
polynomial (with its sign) by each monomial (with its sign) of the
other polynomial. Write these products one after the other (with
their signs) and then add like monomials to form the complete
product.
Example: (x + 3)(2x + 2) = (x + 3)(2x) + (x + 3)2 =
= (2x2 6x) + (2x + 6) = 2x2 4x + 6.
Alberto Pardo Milanes Algebraic expressions

16. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Multiplying polynomials
Multiply it vertically
Sometimes doing it vertically can be nicer:
Example:
(4x2 4x 7)(x + 3) = (4x2 4x 7)x + (4x2 4x 7)3 =
4x3 4x2 7x + 12x2 12x 21 = 4x3 + 8x2 19x 21
4x2 4x 7
x + 3
12x2 12x 21
4x3 4x2 7x
4x3 + 8x2 19x 21
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17. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
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18. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 1
Convert the statements into an algebraic expression using a
variable and a sum or a dierence:
A number plus four:
Five more than a number:
A number minus

19. ve:
The sum of a number and two:
A number increased by ten:
One less than a number:
Seven added to a number:
The dierence of a number and eight:
Nine less than a number
A number decreased by three:
Six subtracted from a number:
The age a boy was two years ago:
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20. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 2
Convert the statements into an algebraic expression using a
variable and a multiplication or a division:
Double a number:
The quotient of a number and six:
The product of four and a number:
Twice a number:
Nine divided by a number:
A number multiplied by negative

21. ve:
One

22. fth of a number:
Three times a number:
The ratio of a number to four:
Eighty percent of a number :
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23. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 3
Write the sentence as an algebraic expression:
My bedroom's lenght is 2 more feet than its width n. The
lenght is . . .
The temperature at noon was t and had risen 8 degrees since
seven o'clock. The temperature at 7:00 was . . .
Lou charges 6;50 euros an hour to baby-sit. Today he worked
x hours which means that he earned . . .
I have y stamps from Asia and I have seven fewer stamps
from Europe than from Asia. The total number of stamps I
have is . . .
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24. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 4
Write the sentence as an algebraic expression and operate:
The base of a rectangle is double than the height. The area of
the rectangle is. . .
The product of a number and the number than comes after it
is. . .
I have nine fewer coins from China than from Australia. The
total number of coins I have is . . .
Tom's age is double than Fred's age. The product of their
ages is. . .
The sum of a number and twice the number that comes
before it is. . .
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25. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 5
Find the degree of these monomials:
Deg(5x4) =
Deg(4y) =
Deg(
1
2
z3) =
Deg(abch2) =
Deg(4xy) =
Deg(x2) =
Deg(33x7) =
Deg(5a4b) =
Deg(
3
5
x2y) =
Deg(3x2y2) =
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26. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 6
Link like monomials:
7x 4x2y
2xy 7xy
3
5
x2y
1
2
x
3xy2 2xy2
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27. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 7
Complete:
Monomial Coecient Order Variables
7ab2
3
2
m3n2p2
7
p
3x3y4
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28. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 8
Find:
2x2 3y = 2x2y3
7
4
x2y = 4xa : 10a =
3
5
x 15y = 5xa 10yb =
9
2
x2y3 :
3
2
xy =
1
2
x2y 3y = 4z 5zy = 9zab2 : 6z =
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29. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 9
Find the degree of these polynomials:
Deg(x4 3x3 + 2x2 + 1) =
Deg(x2 3x3 + 2x + 1) =
Deg(xy x2 + yx3 + y2) =
Deg(x4y2 3x5) =
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30. Index Monomials Operations Polinomials Operations with polynomials Multiplying polynomials Exercises
Exercises
Exercise 10
Find:
(4x3 2x + 5) + (x2 2x + 1) + (4x2 6) =
(7x3 + 2x2 + 4x + 9) (2x3 3x + 8) =
(2x 3) (x + 12) =
(x3 + 3x2 + 3x + 1) (x2 + 1) =
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