Polynomials

Content

• Introduction
• Polynomials
• Polynomial Operations
• Adding and Subtracting Polynomials
• Multiplying Polynomials
• Dividing Polynomials
• Factoring Polynomials
• Factoring out the GCF
• Factoring Trinomials
• Factoring with Reverse FOIL: Trinomials with $a = 1$
• Factoring by Grouping: Trinomials with $a\neq 1$
• Factoring by Grouping: Polynomials with Higher Degrees

Introduction

A polynomial is an expression created by the sum of mathematical terms. Understanding polynomial expressions can help you understand graphs of lines, parabolas, and other functions.

Polynomials

The terms of a polynomial have specific characteristics. See this list to determine what is a term and what is not.

Examples of Terms:

• $3$
• $x$
• $3x^{2}$
• $\frac{1}{2}x$
• $x^{4}$

Example of Not Terms:

• $x^{-2}$
• $\frac{1}{x^{4}}$
• $\sqrt{16}$

A polynomial is made up of more than one term, and the standard form is written by listing the exponents in decreasing order. Here is an example of a polynomial written in the standard form. The labels indicate the numbers that are the coefficients and the constant. This polynomial has a degree of 3, the highest exponent shown, and it is a trinomial because there are three terms: $6x^{3} +4x^{2} +10$.

For terms with more than one variable, add the exponents together to determine the degree: $4x^{2}y^{2}$.

This polynomial has a degree of 7 and is a binomial because there are 2 terms: $2x^{2}y^{5} + x^{3}$.

Polynomial Operations

Just like numbers, polynomials can be added, subtracted, multiplied, and divided. Here are some examples.

Adding and Subtracting Polynomials

To add or subtract polynomials, combine all like terms: $(2x^{2} + 3) + (x^{2} - 10) = 3x^{2} + -7$

Multiplying Polynomials

To multiply polynomials, use the Distributive Property: $(3x) \times (2x^{2}+4x+3) = 6x^{3} + 12x^{2} +9x$

This problem calculates the product of two binomials, a special case. To multiply, use the mnemonic device FOIL (First - Outer - Inner - Last).

• To foil, first multiply the first two terms of each binomial;
• Next, multiply the outer two terms;
• Next, multiply the inner two terms;
• And last, multiply the last two terms:

$(x+2)\times(x+3)=x^{2}+3x+2x+6=x^{2}+5x+6$

Dividing Polynomials

Dividing polynomials is similar to dividing fractions. Like division of numbers, you subtract multiples of the divisor from the dividend as you work through the problem, and it is possible to have a remainder.

Shown here, the operation of division undoes the product from the previous example. The divisor and the quotient are the same as the above binomial factors:

$\begin{align} x+3\\ x+2 ~\overline{\big)x^2+5x+6}\\ \underline{-(x^2 +2x)}\\ 3x+6\\ \underline{-3x+6}\\ 0 \end{align}$

Factoring Polynomials

To factor, set polynomials equal to zero. The factoring of polynomials reveals the roots or zeros of an equation and determines where the graph touches the x-axis at point (x , 0).

Factoring out the GCF

Look for factors common to each of the terms in a polynomial then factor the common factor out of the expression**.

This expression has a common factor of 3x, so by reversing the Distributive Property we move the term outside parentheses and leave what’s left of the expression inside the parentheses. Although this expression is not fully factored, factoring out the GCF makes it easier to complete the factoring process.

$\begin{align} 6x^{3} + 3x^{2} +9x &= 0 \\ 3x(2x^{2} +x +3) &= 0 \end{align}$

Factoring Trinomials

To factor trinomials, there are several different strategies available.

Factoring with Reverse FOIL: Trinomials with $a = 1$

This trinomial is in the standard quadratic form $ax^{2} + bx + c = 0$.

For trinomials with a is equal to 1, factor the trinomial by reversing the FOIL method. A shortcut is to determine the factors of c that will sum to b.

$\begin{align} ax^{2} + bx + c &= 0\\ x^{2} +9x +20&=0\\ (x+4)(x+5) &= 0 \end{align}$

Factoring by Grouping: Trinomials with $a\neq 1$

This trinomial is also in the standard quadratic form $ax^{2} + bx + c = 0$, but a is not equal to 1, so the factoring process is different. Multiply a and c together then determine the factors of that product that will sum to b. Next factor by grouping.

$\begin{align} ax^{2} + bx + c &= 0\\ 2x^{2} +x - 6 &= 0\\ 2x^{2} -3x +4x - 6 &= 0\\ \left(2x^{2}-3x\right) + \left(4x - 6\right) &= 0\\ x(2x -3) +2(2x -3)&=0\\ (x+2)(2x-3) &= 0 \end{align}$

Factoring by Grouping: Polynomials with Higher Degrees

You can also factor polynomials by grouping when the degree is higher than 2.

This polynomial has a degree of 3 and 4 terms.

$\begin{align} x^{3} +2x^{2} -4x -8 &= 0\\ (x^{3} +2x^{2}) + (-4x -8)&=0\\ x^{2}(x+2) + -4(x+2) &= 0\\ (x^{2} -4)(x+6)&=0 \end{align}$