1.5 Factoring polynomials

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In this section students will:
• Factor the greatest common factor of a polynomial.
• Factor a trinomial.
• Factor by grouping.
• Factor a perfect square trinomial.
• Factor a difference of squares.
• Factor the sum and difference of cubes.
• Factor expressions using fractional or negative exponents.

Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in [link] .

The area of the entire region can be found using the formula for the area of a rectangle.

The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of $\text{\hspace{0.17em}}A={s}^{2}={4}^{2}=16\text{\hspace{0.17em}}$ units 2 . The other rectangular region has one side of length $\text{\hspace{0.17em}}10x-8\text{\hspace{0.17em}}$ and one side of length $\text{\hspace{0.17em}}4,$ giving an area of $\text{\hspace{0.17em}}A=lw=4\left(10x-8\right)=40x-32\text{\hspace{0.17em}}$ units 2 . So the region that must be subtracted has an area of $\text{\hspace{0.17em}}2\left(16\right)+40x-32=40x\text{\hspace{0.17em}}$ units 2 .

The area of the region that requires grass seed is found by subtracting $\text{\hspace{0.17em}}60{x}^{2}-40x\text{\hspace{0.17em}}$ units 2 . This area can also be expressed in factored form as $\text{\hspace{0.17em}}20x\left(3x-2\right)\text{\hspace{0.17em}}$ units 2 . We can confirm that this is an equivalent expression by multiplying.

Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

Factoring the greatest common factor of a polynomial

When we study fractions, we learn that the greatest common factor    (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, $\text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ is the GCF of $\text{\hspace{0.17em}}16\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ because it is the largest number that divides evenly into both $\text{\hspace{0.17em}}16\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ The GCF of polynomials works the same way: $\text{\hspace{0.17em}}4x\text{\hspace{0.17em}}$ is the GCF of $\text{\hspace{0.17em}}16x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}20{x}^{2}\text{\hspace{0.17em}}$ because it is the largest polynomial that divides evenly into both $\text{\hspace{0.17em}}16x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}20{x}^{2}.$

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

Greatest common factor

The greatest common factor    (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

Given a polynomial expression, factor out the greatest common factor.

1. Identify the GCF of the coefficients.
2. Identify the GCF of the variables.
3. Combine to find the GCF of the expression.
4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.
5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

Factoring the greatest common factor

Factor $\text{\hspace{0.17em}}6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy.$

First, find the GCF of the expression. The GCF of $\text{\hspace{0.17em}}6,45,$ and $\text{\hspace{0.17em}}21\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}3.\text{\hspace{0.17em}}$ The GCF of $\text{\hspace{0.17em}}{x}^{3},{x}^{2},$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ (Note that the GCF of a set of expressions in the form $\text{\hspace{0.17em}}{x}^{n}\text{\hspace{0.17em}}$ will always be the exponent of lowest degree.) And the GCF of $\text{\hspace{0.17em}}{y}^{3},{y}^{2},$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ Combine these to find the GCF of the polynomial, $\text{\hspace{0.17em}}3xy.$

Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that $\text{\hspace{0.17em}}3xy\left(2{x}^{2}{y}^{2}\right)=6{x}^{3}{y}^{3},3xy\left(15xy\right)=45{x}^{2}{y}^{2},$ and $\text{\hspace{0.17em}}3xy\left(7\right)=21xy.$

Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.

$\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right)$

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