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

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.

𝑃 (𝑥 )=𝑎0+𝑎1𝑥+𝑎2𝑥2+𝑎3𝑥

3+…+𝑎𝑛 𝑥𝑛

where “x” is a variable.“” are real numbers, the coefficients“n” is the degree of the polynomial.

For example:

is a 4th degree polynomial.

is a 6th degree polynomial.

1.1. Polynomials.

1. Polynomial roots.

A monomial is an only one term polynomial. 𝑃 (𝑥 )=−6 𝑥2

A binomial has two elements; 𝑃 (𝑥 )=4 𝑥−5 𝑥3

We can change the x for one number and in that case we can obtain the value of the polynomial;

𝑃 (𝑥 )=−3−5 𝑥2−4 𝑥3+10 𝑥6

…

1.1. Polynomials.

1. Polynomials.

1.2. Polynomial roots.

A root of a polynomial is a number “x1”, such that . The fundamental theorem of algebra states that a polynomial of degree n, has n roots, some fo which may be degenerate.

For instance, if we have the next polynomial;

If , = 0, so that is a root of the polynomial.

If = 0, so that -6 is another root of the polynomial.

1. Polynomials.

1.2. Polynomial factorization.

Finding roots of a polynomial is therefore equivalent to polynomial factorization into factors of degree 1.

For example;

But how can we factorize the polynomials? We can distinguish 4 cases;

a) Second degree polynomials:In these cases we have to resolve the equation.

; we resolve the equation , and the solutions are and . So we can factorize the

polynomial in this way:

1. Polynomials.

1.2. Polynomial factorization.

b) Biquadratic polynomials.In these cases we have to do a variable change; , so we can obtain a

second degree equation;

+1 ; changing the variable, , we obtain the polynomial, .

As it is a second degree polynomial, we can resolve the equation . It has a unique solution (that

appears twice), .

So, , we have two double solutions

1. Polynomials.

1.2. Polynomial factorization.

c) The cases where we have “x” as a common factor.In these cases we can take out the common factor and the degree of

the polynomial decreases in one unit;

; we can take out the common factor “x”;

And the we can resolve the second degree equation and factorize the polynomial;

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;

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;

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;

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.

2. Logarithm.

2. Logarithm.

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.

2.1. Logarithm's properties.

3. Equations.

3.1. Exponential equations.

3.2. Logarithmic equations.

3.2. Nonlinear equation system.

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.

xxx

632

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.

3. Equations.

3.4. Inequations.

3. Equations.

3.4.1. Linear Inequations of one unique unknown.

They have this form:, or , or , or .

They can be resolved as in the example before.

3. Equations.

3.4.2. Second degree Inequations of one unique unknown.

They have this form:.

To resolve these inequations, we have to resolve the equation first, and then we will obtain two solutions; and . So that the real line is divided in three intervals.

To resolve the inequiation, we have to choose one number of each interval and replace them in the inequation, to see if it satisfy it or not.

The limit points are going to be of the solution, depending on the symbol that we have in the inequation;

Example:

3. Equations.

3.4.3. Linear inequations systems of one unique unknown.

The solution of the linear inequation system of one unique unknown, is going to be the intersection of the solutions of all the inequations of the system.

Example:

The solution of the first inequation is .The solution of the second inequation is

So the solution to the system is , which is

3. Equations.

3.4.3. Linear inequations systems of one unique unknown.

Be careful!!

If the inequation has a fraction, and the unknown is in the numerator, we can not simplify the numerator after doing the g.c.f. (greater common factor), because we don’t know if the numerator has a positive or negative sign.

What an we do? After doing the g.c.f. and pass all the terms to one side of the inequation, we have to think about all possibilities of the division.

Example: ;