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6. Polynomials & Factoring
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Algebra 1

6.06 Factor trinomials

Lesson
Practice

Factor trinomials

Trinomials can be rewritten as polynomials with four terms and factored by grouping.

Exploration

Consider the polynomial expressions factored by grouping below:

Twelve expressions arranged into 3 columns with 4 rows in each column. First column: 3 x squared plus 7 x plus 2, 3 x squared plus 6 x plus 1 x plus 2, 3 x left parenthesis x plus 2 right parenthesis plus 1 left parenthesis x plus 2 right parenthesis, left parenthesis x plus 2 right parenthesis left parenthesis 3 x plus 1 right parenthesis; Second column: 2 x squared plus 11 x plus 12, 2 x squared plus 8 x plus 3 x plus 12, 2 x left parenthesis x plus 4 right parenthesis plus 3 left parenthesis x plus 4 right parenthesis, left parenthesis x plus 4 right parenthesis left parenthesis 2 x plus 3 right parenthesis; Third column: x squared plus 5 x plus 6, x squared plus 3 x plus 2 x plus 6, x left parenthesis x plus 3 right parenthesis plus 2 left parenthesis x plus 3 right parenthesis, left parenthesis x plus 3 right parenthesis left parenthesis x plus 2 right parenthesis.
  1. What patterns do you notice between the original expression and the terms used to rewrite the linear term?
  2. Choose one of the linear terms and rewrite the term in a different way than shown, then determine whether the polynomial can still be factored by grouping.

When using the grouping method to factor a trinomial we must first rewrite the trinomial as a polynomial with 4 terms so that we can make 2 groups. The linear \left(x \right) terms we use to rewrite the original polynomial must add to give the same linear term as the original trinomial. Their coefficients also need to multiply to the same number that the original trinomial's leading coefficient and constant multiply to.

In other words if the trinomial ax^2+bx+c is rewritten as ax^2+rx+sx+c then r+s=b and {r \cdot s=a \cdot c}.

Steps in factoring a quadratic trinomial of the form ax^{2} + bx + c:

  1. Factor out any GCF.

    (If a is negative, we can also divide out a factor of -1 before continuing.)

  2. Find two numbers, r and s, that multiply to ac and add to b.

  3. Rewrite the trinomial with four terms in the form ax^{2} + rx + sx + c.

  4. Factor by grouping.

  5. Check whether the answer will not factor further and verify the factored form by multiplication.

Remember to include any common factors divided out at the start, so each step results in an equivalent expression.

Algebra tiles can also be useful in factoring. Consider the expression 3x^2 + 7x - 6 as the area of a rectangle. If we can find the lengths of this rectangle, then we will have two expressions that multiply to 3x^2 + 7x - 6 because the area of a rectangle is A=l \cdot w.

The image show an area model using algebra tiles: +x squared tiles, +x tiles, and -1 tiles. Ask your teacher for more information.

We don't yet know the side lengths of the rectangle, but we will take 3 of the x^2 tiles, 7 of the +x tiles, and 6 of the -1 tiles and arrange them as closely into a rectangle as we can.

We will start by lining up all of the x^2 tiles, then put the x tiles underneath to match the equal lengths. Finally, put the -1 tiles next to the x tiles to match the equal lengths.

Notice we have some empty spaces that need to be filled in.

The image show area models using algebra tiles: +x squared tiles, +x tiles, -x tiles, and -1 tiles. Ask your teacher for more information.

Notice that x tiles will fit perfectly into the empty spaces. However, we don't want to change the value of the expression so we need to make sure to add zero pairs.

A zero pair is two values that add to 0. x and -x is a zero pair.

Since there are 4 empty spaces for x tiles, we can fill 2 spaces with (positive) x tiles and 2 spaces with -x tiles.

Technically this represents the expression 3x^2+9x-2x-6 which is equivalent to 3x^2+7x-6 by combining like terms.

The image show area models using algebra tiles: +x squared tiles, +x tiles, -x tiles, +1 tiles, and -1 tiles. Ask your teacher for more information.

Now we can use the lengths of the sides of the rectangle to determine the expressions that can be multiplied together to create the original expression 3x^2+7x-6.

The x^2 tile has side lengths of x and x. The x tiles have a shorter side length of 1 and a longer side length of x. The -x tiles have a shorter side length of -1 and a longer side length of x.

The shorter side length of the rectangle is x+3 units and the longer side length is 3x-2 units.

This shows us: 3x^2 + 7x - 6 = 3x^2 + 9x -2x + 6= (x+3)(3x-2).

Examples

Example 1

Factor x^{2} + 10 x - 24.

Worked Solution
Create a strategy

Since there are no common factors for all three terms, we proceed with finding the value of two integers that multiply to ac = (1)(-24) = -24 and add up to b = 10. After finding these integers, we use them to rewrite the middle term 10x as a sum of two terms and then factor the trinomial by grouping.

Apply the idea

The factors of -24 are 1 and -24,\,-1 and 24,\,2 and -12,\,-2 and 12,\,3 and -8,\,-3 and 8,\,4 and -6,\,-4 and 6. Among these factors, -2 and 12 are the pair that add up to 10.

We can use this to rewrite the trinomial and factor by grouping as follows:

\displaystyle x^{2} + 10 x - 24\displaystyle =\displaystyle x^{2} + 12x - 2x - 24Rewrite polynomial with four terms
\displaystyle =\displaystyle x(x+12) -2(x+12)Factor each pair
\displaystyle =\displaystyle (x+12)(x-2)Divide out common factor of (x+12)

There are no more common factors to be divided out, so the fully factored form of the polynomial is (x+12)(x-2).

Reflect and check

We can perform a midway check that we are factoring by grouping appropriately when we factor out a GCF from each set of binomials in the step x(x+12) -2(x+12).

If we factor out a GCF at this step and the binomial factors are not equivalent, we may have split the linear term from x^{2} + 10 x - 24 incorrectly or factored out a GCF incorrectly. This is an important place to stop and check that we are factoring appropriately.

Also note that we could have also rewritten the polynomial as x^2-2x+12x-24. This would have resulted in a different middle step in factoring by grouping but the same end result.

Example 2

Factor 3 x^{2} - 27.

Worked Solution
Create a strategy

We can factor a GCF of 3 out of the polynomial and write the polynomial as 3(x^2-9). Since the linear term is missing from the polynomial, we can write the polynomial as 3(x^2+0x-9). There are no common factors. We will find the value of two integers that multiply to ac=(1)(-9)=-9 and add up to b=0. After finding these integers, we use them to rewrite the middle term 0x as a sum of two terms. Then factor the trinomial by grouping.

Apply the idea

The factors of -9 are 1 and -9,\,-1 and 9,\,3 and -3. Among these factors, 3 and -3 are the pair that add up to 0.

We can use this to rewrite the trinomial and factor by grouping as follows:

\displaystyle 3(x^{2} + 0x - 9)\displaystyle =\displaystyle 3(x^{2} + 3x - 3x - 9)Rewrite polynomial with four terms
\displaystyle =\displaystyle 3[x(x+3)-3(x+3)]Factor each pair
\displaystyle =\displaystyle 3(x+3)(x-3)Divide out common factor of (x+3)

There are no more common factors to be divided out, so the fully factored form of the polynomial is 3(x+3)(x-3).

Reflect and check

Recall that the special product of a sum and difference gives us: \left(a+b\right)\left(a-b\right) = a^{2} - b^{2}. Notice that the factored form of the binomial x^2-9=(x+3)(x-3).

This can also be verified using algebra tiles:

The image show an area model using algebra tiles with key: +x squared tiles, +x tiles, -x tiles, +1 tiles, and -1 tiles. Ask your teacher for more information.

Notice the x terms form a total of 0. So, we know:

3x^2 - 27 = (3x+9)(x-3) = 3(x+3)(x-3)

Example 3

Factor 5 x^{2} - 18x + 9.

Worked Solution
Create a strategy

Since there are no common factors for all three terms, we proceed with finding the value of two integers that multiply to ac = 5 \cdot 9 = 45 and add up to b = -18. After finding these integers, we use them to rewrite the middle term -18x as a sum of two terms and then factor the trinomial by grouping.

Apply the idea

The factor pairs of 45 are 1 and 45,\,-1 and -45,\,3 and 15,\,-3 and -15,\,5 and 9,\,-5 and -9. Note that since the middle term of the trinomial is negative, we need to consider negative and positive factors. Of these factors, -15 and -3 are the pair that adds up to -18.

We can use this to rewrite the trinomial and factor by grouping as follows:

\displaystyle 5x^2 - 18x + 9\displaystyle =\displaystyle 5x^2 - 15x - 3x + 9Rewrite polynomial with four terms
\displaystyle =\displaystyle 5x\left(x - 3\right) - 3\left(x - 3\right)Factor each pair to leave behind a common binomial
\displaystyle =\displaystyle \left(x - 3\right)\left(5x - 3\right)Divide out the common factor of \left(x - 3\right)

There are no more factors to be taken out, so the fully factored form of the polynomial is \left(x - 3\right)\left(5x - 3\right).

Reflect and check

We can check the answer by multiplying the factored form \left(5x-3\right)\left(x-3\right).

A rectangle divided into 2 rows of 2 rectangles. From top to bottom, the left column of rectangles is labeled 5 x squared and negative 15 x, and the right column of rectangles is labeled negative 3 x and 9. The outside of the rectangle is labeled 5 x above the left column, 3 above the right column, and a minus sign above the line between the left and right column. The side is labeled x next to the top row and negative 3 next to the bottom row.

The polynomial 5x^2-3x-15x+9 simplifies to 5x^2-18x+9.

Idea summary

Steps in factoring a quadratic trinomial:

  1. Factor out any GCF.

    (If a is negative, we can also divide out a factor of -1 before continuing.)

  2. Find two numbers, r and s, that multiply to ac and add to b.

  3. Rewrite the trinomial with four terms, in the form ax^{2} + rx + sx + c.

  4. Factor by grouping.

  5. Check whether the answer will not factor further and verify the factored form by multiplication.

Remember to include any common factors divided out at the start, so each step results in an equivalent expression.

Outcomes

A.EO.2

The student will perform operations on and factor polynomial expressions in one variable.

A.EO.2c

Factor completely first- and second-degree polynomials in one variable with integral coefficients. After factoring out the greatest common factor (GCF), leading coefficients should have no more than four factors.

A.EO.2e

Represent and demonstrate equality of quadratic expressions in different forms (e.g., concrete, verbal, symbolic, and graphical).

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