## Key Concepts

- Factor a quadratic trinomial when a is not equal to 1
- Factor out a Common Factor
- Understand factoring by grouping
- Factor a trinomial using substitution

### Factoring a trinomial when a is not equal to 1

#### Factor out a Common Factor

**What is the factored form of 3x**^{3}** + 15x**^{2}** – 18x?**

Before factoring the trinomial into two binomials, look for any common factors that you can factor out.

So, 3x^{3} + 15x^{2} – 18x = 3x(x^{2} + 5x – 6).

Then factor the resulting trinomial, x^{2} + 5x – 6.

The factored form of x^{2} + 5x – 6 is (x – 1)(x + 6), so the factored form of 3x^{3} + 15x^{2} – 18x is 3x(x – 1)(x + 6).

### Understand factoring by grouping

**If ax**^{2}+ bx + c is a product of binomials, how are the values of a, b and c related? Consider the product (3x + 4)(2x + 1).

The product is 6x^{2} + 11x + 4. Notice that ac = (6)(4) or (3)(2)(4)(1), which is the product of all of the coefficients and constants from (3x + 4)(2x + 1).

In the middle terms, the coefficients of the x-terms, 3 and 8, add to form b = 11. They are composed of the pairs of the coefficients and constants from the original product; 3 = (3)(1) and 8 = (4)(2).

If ax^{2} + bx + c is the product of the binomials, there is a pair of the factors of ac that have a sum of b.

**How can you factor ax**^{2}**+ bx + c by grouping?**

Consider the trinomial 6x^{2} + 11x + 4, a = 6 and c = 4, so ac = 24.

Find the factor pair of 24 with a sum of 11

The factored form of 6x^{2} + 11x + 4 is (3x + 4)(2x + 1).

**Check. **(3x + 4)(2x + 1) = 6x^{2} + 3x + 8x + 4 = 6x^{2} + 11x + 4

**Factoring a trinomial using substitution method**

How can you use substitution to help you factor ax^{2} + bx + c as the product of two binomials?

Consider the trinomial 3x^{2} – 2x – 8.

**Step 1. **Multiply ax^{2} + bx + c by a to transform x^{2} into (ax)^{2}.

**Step 2.** Replace ax with a single variable. Let p = ax.

= p^{2} – 2p – 24

**Step 3.** Factor the trinomial.

= (p – 6)(p + 4)

**Step 4. **

Substitute ax back into the product. Remember p = 3x. Factor out the common factors if there are any.

**Step 5. **

Since you started by multiplying the trinomial by a, you must divide by a to get a product that is equivalent to original trinomial.

The factored form of 3x^{2} – 2x – 8 is (x – 2)(3x + 4). In general, you can use substitution to help transform ax^{2} + bx + c with a not equal to 1 to a simpler case in which a = 1, factor it, and then transform it back to an equivalent factored form.

### Questions

**Question 1**

Write the factored form of each trinomial.

1. **5x ^{2} – 35x + 50 **

Take out 5.

5(x^{2} – 7x + 10)

Find the factors of x^{2} – 7x + 10

Factors of 10 are -5 and -2.

-5 + (-2) = -7

So, (x – 5)(x – 2)

Ans: 5(x – 5)(x – 2)

2. **6x ^{3} + 30x^{2} + 24x **

Take out 6x.

6x(x^{2} + 5x + 4)

Find the factors of x^{2} + 5x + 4

Factors of 4 are 4 and 1.

4 + 1 = 5

So, (x + 4)(x + 1)

Ans: 6x(x + 4)(x + 1)

3. **10x ^{2} + 17x + 3 **

a = 10, b = 17, c = 3

a × c = 10 × 3 = 30

Factors of 30 are 15 and 2.

And 15 + 2 = 17

10x^{2} + 15x + 2x + 3

= 5x(2x + 3) + 1(2x + 3)

= (5x + 1)(2x + 3)

4. **2x ^{2} – x – 6 **

a = 2, b = -1, c = -6

a × c = -12

Factors of -12 are -4 and 3.

And (-4) + 3 = -1

2x^{2} – 4x + 3x – 6

= 2x(x – 2) + 3(x – 2)

= (2x + 3)(x – 2)

5. **10x ^{2} + 3x – 1 **

a = 10, b = 3, c = -1

a*c = -10

Factors of -10 are 5 and -2.

And 5 + (-2) = 3

10x^{2} + 5x – 2x – 1

= 5x(2x + 1) + (-1)(2x + 1)

= (5x – 1)(2x + 1)

**Question 2**

A photographer is placing photos in a mat for a gallery show. Each mat she uses is x in. wide on each side. The total area of each photo and mat is shown below.

- Factor the total area to find the possible dimensions of a photo and mat.

- What are the dimensions of the photos in terms of x?

**Solution: **

**1. Total area of the photo and the mat = 4x^2 + 36x + 80 **

Let’s factor this trinomial using the substitution method.

(2x)^{ }2 + 18(2x) + 80

Let 2x = p.

The trinomial becomes p^{2} + 18p + 80.

Factors of 80 are 10 and 8.

And 10 + 8 = 18

(p + 10)(p + 8)

As p = 2x,

4x^{2} + 36x + 80 = (2x + 10)(2x + 8)

**2. The dimensions of the photo and the mat combined are 2x + 10 in. by 2x + 8 in. **

To find the dimension of just the photo, subtract 2x from both length and width.

L = (2x + 10) – 2x = 10 in.

W = (2x + 8) – 2x = 8 in.

The dimensions of each photo are 10 in. by 8 in. In terms of x, it is 10x^{0} in. by 8x^{0} in. (independent of the value of x).

### Key Concepts Covered

## Exercise

Factor the following trinomials:

- 8x
^{2}– 10x – 3 - 12x
^{2}+ 16x + 5 - 16x
^{3}+ 32x^{2}+ 12x - 21x
^{2}– 35x – 14 - 16x
^{2}+ 22x – 3 - 9x
^{2}+ 46x + 5 - –6x
^{2}– 25x – 25 - 5x
^{2}– 4xy – y^{2} - 16x
^{2}+ 60x – 100 - 6x
^{2}+ 5x – 6

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