# Factoring Trinomials Worksheet

February 2, 2023

A trinomial is a polynomial with three terms. It commonly takes the shape of \(x^2 + bx + c\). Factoring trinomials can be challenging, but this worksheet for factoring trinomials with an answer key can be helpful for students to check their work and understand the concepts.

## Factoring Trinomials: \(x^2 + bx + c\)

Trinomials in the form \(x^2 + bx + c\) can frequently be factored as the result of multiplying two binomials. A binomial is just a polynomial with two terms. Let’s examine the outcome when we multiply two binomials, such as (x + 2) and (x + 3).

### Example

Multiply (x + 2)(x + 3).

(x + 2)(x + 3)

Use the FOIL method to multiply binomials.

\( x^2 + (2 + 3)x + (2 × 3) \)

Answer: \( x^2 + 5x + 6 \)

## Factoring trinomials worksheet (a = 1)

Write each trinomial in factored form.

- \(x^2 + 5x + 6\)
- \(x^2 – 7x + 10\)
- \(x^2 + 8x + 15\)
- \(x^2 – 9x + 18\)
- \(x^2 + 4x + 4\)

## Factoring Trinomials Worksheet Answers with Solutions

**Question 1: \(x^2 + 5x + 6\)**

Identify the values for b and c.

Here, b = 5 and c = 6.

We have to find two numbers that add to b and Multiply to c.

This step can take a little bit of trial and error.

For instance, you could pick 6 and 1 because \( 6 × 1 = 6 \). But \( 6 + 1 \) does not equal 5, so these numbers would not work.

However, if you chose 3 and 2, you can easily confirm that:

\( 3 + 2 = 5 \) (the value of b); and

\( 3 × 2 = 6 \) (the value of c)

So, the factors would be (\(x + 2\)) and (\(x + 3\))

**Question 2: \(x^2 – 7x + 10\)**

Here, b = 7 and c = 10.

We have to find two numbers that add to b and Multiply to c.

\( 5 + 2 = 7 \) (the value of b); and

\( 5 × 2 = 10 \) (the value of c)

So, the factors would be (\( x – 5 \)) and (\( x-2 \))

**Question 3: \(x^2 + 8x + 15\)**

Here, b = 8 and c = 15.

We have to find two numbers that add to b and Multiply to c.

\( 5 + 3 = 8 \) (the value of b); and

\( 5 × 3 = 15 \) (the value of c)

So, the factors would be (\( x + 3 \)) and (\( x + 5 \))

**Question 4: \(x^2 – 9x + 18\)**

Here, b = 9 and c = 18.

We have to find two numbers that add to b and Multiply to c.

\( 6 + 3 = 9 \) (the value of b); and

\( 6 × 3 = 18 \) (the value of c)

So, the factors would be (\( x – 6 \)) and (\( x – 3 \))

**Question 5: \(x^2 + 4x + 4\)**

Here, b = 4 and c = 4.

\( 2 + 2 = 4 \) (the value of b); and

\( 2 × 2 = 4 \) (the value of c)

So, the factors would be (\( x + 2 \)) and (\( x + 2 \))

## Factoring Perfect Square Trinomials

Some polynomials have common patterns, which can be factorized faster if you recognize them.

## How to factor perfect square trinomials

A trinomial is a perfect square if its first and last terms are both perfect squares, and the middle term can be derived by multiplying the first and last terms, then multiplying the result by 2. To determine if a trinomial is a perfect square, we check if its first and last terms are both squares. In the example \( x^2 + 8x + 16 \), both \( x^2 \) and 16 are squares, and the middle term 8x can be derived from \(2 × x × 4 \), making it a perfect square trinomial. We can rewrite it as \( (x + 4)^2 \) by factoring.

## Factoring Perfect Square Trinomials Worksheet

- \( x^2 + 6x + 9 \)
- \( y^2 – 10y + 25 \)
- \( a^2 + 2ab + b^2 \)
- \( z^2 – 14z + 49 \)
- \( m^2 + 4m + 4 \)

## Factoring Perfect Square Trinomials Worksheet Answers with Solutions

**Question 1: \( x^2 + 6x + 9 \)**

We know this is a perfect square. Therefore, look at the first and last term and find what they are squares of. It’ll give us: \((x + 3)^2\)

**Question 2: \( y^2 – 10y + 25 \)**

We know that \( 5^2 = 25 \)

So the factor will be \((y – 5)^2\)

**Question 3: \( a^2 + 2ab + b^2 \)**

Here \( b^2 \)is the perfect square, so factoring it we will get

\( = (a + b)^2\)

**Question 4: \( z^2 – 14z + 49 \)**

\( 7^2 = 49 \)

\( = (z – 7)^2\)

**Question 5: \( m^2 + 4m + 4 \)**

\( 2^2 = 4 \)

\( = (m + 2)^2\)

## Factoring Tips

Factoring trinomials requires practice and patience. Sometimes the correct number combinations become apparent, but other times finding the right ones can be difficult despite trying various options. Additionally, some trinomials cannot be factored in. Although there is no guaranteed method to get the right answer immediately, some helpful tips can make the process easier.

## Tips for Finding Values that Work** **

Use the following tips when factoring a trinomial in form \( x^2 + bx + c \)

**Start by examining the c term**

- If c is positive, the factors of c will be positive or negative, meaning that r and s will have the same sign.
- If c is negative, one factor will be positive, and the other will be negative, resulting in either r or s being negative but not both.

**Next, consider the b term**

- If c is positive and b is positive, both r and s will be positive.
- If c is positive and b is negative, both r and s will be negative.
- If c is negative and b is positive, the factor with the larger absolute value will be positive (e.g., if |r| > |s|, then r is positive and s is negative).
- If c is negative and b is negative, the factor with the larger absolute value will be negative (e.g., if |r| > |s|, then r is negative and s is positive).

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