DU2 Polynomials, logarithms, exponentials, and inequalities

1. DU2: Polynomials, logarithms,
exponentials, and inequalities.
mathematics

2. 1. Polynomials.
1.1. Polynomials.
A polynomial is an expression that is the sum of a finite number of non-zero terms,
each term consisting of the product of a constant and a finite number of variables
raised to whole number powers.

3. 1. Polynomial roots.
1.1. Polynomials.
A monomial is an only one term polynomial.
A binomial has two elements;
We can change the x for one number and in that case we can obtain the value of the
polynomial;

4. 1. Polynomials.
1.2. Polynomial roots.

5. 1. Polynomials.
1.2. Polynomial factorization.

6. 1. Polynomials.
1.2. Polynomial factorization.

7. 1. Polynomials.
1.2. Polynomial factorization.

8. 1. Polynomials.
1.2. Polynomial factorization.
d) When the polynomial is 3th degree or more:
In these cases we have to use the RUFINNI’S RULE;

9. 1. Polynomials.
1.2. Polynomial factorization.
d) When the polynomial is 3th degree or more:
In these cases we have to use the RUFINNI’S RULE;

10. 1. Polynomials.
1.2. Polynomial factorization.
d) When the polynomial is 3th degree or more:
In these cases we have to use the RUFINNI’S RULE;

11. 2. Logarithm.
The logarithm of a number is the exponent to which another fixed value, the base,
must be raised to produce that number.
For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power
3: 1000 = 10 × 10 × 10 = 103.
More generally, if x = by, then y is the logarithm of x to base b, and is written y =
logb(x), so log10(1000) = 3.

12. 2. Logarithm.

13. 2. Logarithm.
2.1. Logarithm's properties.
The logarithm of a multiplication:
Loga (x . y) = Loga x + Loga y
The logarithm of a division :
Loga (x / y) = Loga x - Loga y
The logarithm of an exponential:
Loga xb = b . Loga x
Changing the base:
.
Log a N = Log N / Log a

14. 3. Equations.
3.1. Exponential equations.
3.2. Logarithmic equations.
3.2. Nonlinear equation system.

15. 3. Equations.
3.4. Inequations.
In mathematics, an inequation is a statement that an inequality holds between two
values. It is usually written in the form of a pair of expressions denoting the values in
question, with a relational sign between them indicating the specific inequality relation.
x
3x 6 x
2
We can operate with the inequations;
• If we add or rest a number in both parts of the inequation, then the inequation that
we obtain is equivalent to the previous one.
• If we multiplicate or divide with the same number both parts of the inequation, then
the inequation that we obtain is equivalent to the previous one.
• If we multiplicate or divide with the same negative real number both parts of the
inequation, then the inequation will change the sign of the inequality.

16. 3. Equations.
3.4. Inequations.

17. 3. Equations.
3.4.1. Linear Inequations of one unique unknown.

18. 3. Equations.
3.4.2. Second degree Inequations of one unique unknown.

19. 3. Equations.
3.4.3. Linear inequations systems of one unique unknown.

20. 3. Equations.
3.4.3. Linear inequations systems of one unique unknown.