## Posts Tagged ‘polynomial’

## Divide polynomial functions.

**Divide polynomial functions.**

**Dividing Functions**

If *f *(*x*) and *g *(*x*) define functions, then

** .**Quotient function

The domain of the quotient function is the intersection of the domains

of *f *(*x*) and *g *(*x*), excluding any values of *x *for which *g *(*x*) = 0.

## Polynomial Function

**Polynomial Function**

A **polynomial function of degree **** n **is defined by

f(x) = a_{n}x^{n }+ a_{n-1} x^{n-1} + ….. +a_{1}x + a_{0,..}

for real numbers a_{n}, a_{n-1},…..a_{1}, and a_{0} , where a_{n} ≠ 0 and n is a whole number.

## Polynomial

**Polynomial**

A **polynomial **is a term or a finite sum of terms in which all variables

have whole number exponents and no variables appear in denominators

or under radicals.

Examples of polynomials include

3x – 5 , 4m^{3} – ^{5m2p} + 8

## Quadratic and Other Polynomial Inequalities

**Quadratic and Other Polynomial Inequalities**

Inequalities like the following are called quadratic inequalities:

x^{2} + 3x – 10 < 0, 5x^{2} – 3x + 2 ≥ 0.

In each case, we have a polynomial of degree 2 on the left. We will solve such inequalities in two ways. The first method provides understanding and the second yields the more efficient method.

The first method for solving a quadratic inequality, such as ax^{2} + bx + c > 0, is by considering the graph of a related function, f(x) = ax^{2} + bx + c.

**To solve a polynomial inequality:**

**a. **Get 0 on one side, set the expression on the other side equal to 0,

and solve to find the *x*-intercepts.

**b. **Use the numbers found in step (a) to divide the number line into

intervals.

**c. **Substitute a number from each interval into the related function.

If the function value is positive, then the expression will be

positive for all numbers in the interval. If the function value is

negative, then the expression will be negative for all numbers in

the interval.

**d. **Select the intervals for which the inequality is satisfied and write

set-builder or interval notation for the solution set.

## Factoring When Terms Have a Common Factor

**Factoring When Terms Have a Common Factor**

The polynomials we consider most when factoring are those with more than one term. To multiply a monomial and a polynomial with more than one term, we multiply each term of the polynomial by the monomial using the

distributive laws:

*a(b + c) = ab + ac and a(b – c)= ab – ac.*

** **

To factor, we do the reverse. We express a polynomial as a product using the distributive laws in reverse:

** ab **+

**=**

*ac*

*a(b +*

*c) and*

*ab –*

*ac =*

*a(b –*

*c)***.**

** **

**TIPS FOR FACTORING**

**a. **Before doing any other kind of factoring, first try to factor out the GCE.

**b. **Always check the result of factoring by multiplying.

## Finding the Greatest Common Factor

**Finding the Greatest Common Factor**

The numbers 20 and 30 have several factors in common, among them 2 and 5. The greatest of the common factors is called the **greatest common factor, GCF. **One way to find the GCF is by making a list of factors of each number.

List all the factors of 20: 1, 2, 4, 5, 10, and 20.

List all the factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30.

Now list the numbers common to both lists, the common factors: 1, 2, 5, and 10.

Then the greatest common factor, the GCF, is 10, the largest number in the common list.

**IMPORTANT POINTS OF GREATEST COMMOM FACTOR**

**FACTOR AND FACTORIZATION**

**a.** To **factor **a polynomial is to express it as a product.

**b.** A **factor **of a polynomial *P *is a polynomial that can be used to express *P *as a product.

**c.** A **factorization **of a polynomial is an expression that names that polynomial as a product.

**TO FIND THE GCF OF TWO OR MORE MONOMIALS**

** **

**a.** Find the prime factorization of the coefficients, including _1 as a factor if any coefficient is negative.

**b. **Determine any common prime factors of the coefficients. For each one that occurs, include it as a factor of the GCF. If none occurs, use 1 as a factor.

**c. **Examine each of the variables as factors. If any appear as a factor of all the monomials, include it as a factor, using the smallest exponent of the variable. If none occurs in all the monomials, use 1 as a factor.

**d. **The GCF is the product of the results of steps (b) and (c).