Factoring Quadratics by Grouping

Factoring Quadratics by Grouping

What is factoring by grouping?

• A way to factor polynomials by grouping terms together (associative property) and finding common factors (distributive property)

• Factoring by grouping separates the middle term of the quadratic into the two numbers that formed its sum

What is factoring by grouping?

• Reverse of distributive property (factor out common terms)

• Reverse of multiplying polynomials (rethink how we got each term)

• Reverse of combining like terms (separate like terms into the sum of like terms)

• What that means is that we work backwards from multiplying polynomials

Let's take a look at how we get the middle term when we multiply binomials.

• The middle term is the SUM of the products of the INSIDE terms and OUTSIDE terms

• In the example that is 9x + 1x

• 9x + 1x = 10x

• 
  

Steps to Factor by Grouping

• Multiply the coefficient of the first term and the last term together

• List all the factors of that product.

• Find the two factors whose sum is the middle term

• Rewrite the trinomial with the middle term split into the sum of the factors

• Group terms together by twos

• Factor out common term

Example 1

• 1. Multiply coefficient of the first term times the last term: 2 * (-15) = -30

• 2. List all the factors of -30 until you find the two factors that add up to the middle term.

• Middle term is -7. (-10) * (3) =-30. (-10) + (7) = -3

Example 1

• Rewrite the quadratic, but write the middle term as the sum of the two factors of -30

• Group the terms in two groups of two: (2x² - 10x) + (3x - 15)

• Factor out any common terms in each group. If the only common factor is one, use 1. 2x (x-5) + 3(x-5)

• Factor out any factors common to both groups.

• (x-5) is in both groups: (x-5)(2x+3)

Example 2

• Multiply leading coefficient and last term: 3 * 10 = 30

• List all the factors of 30. Remember if the last term is +, then either both factors are +, or both factors are -.

• Find the factors whose sum is the coefficient of the middle term = 11x: 5, 6

• Rewrite the quadratic equation with the middle term written as the sum of the factors: 3x² + 6x + 5x + 10

Example 2

• Group in two groups of 2: (3x² + 6x) + (5x+10)

• Factor out common terms in each group: 3x(x + 2) + 5(x+2)

• Notice (x+2) appears in both groups

• Factor out the factor common to both groups. What remains is the other factor: (x+2)(3x + 5)

• Final answer: (x+2)(3x + 5)

How to organize your factors and sums

• The more organized you can be with your factors and sums, the easier it will be to decide which factors form the sum you need

• One way to do that is the X or diamond method.

• Put the product of the coefficient of the first term and the last term at the top

• Put the coefficient of the middle term at the bottom

• Find the factors of the top number that add to the bottom number

Now You Try

5x² - 2x - 3

What is the first step in factoring by grouping?

Multiple Choice

5x² - 2x - 3

What is the first step in factoring by grouping?

• Find the factors of negative -3.

• Multiply (5) * (-3)

• Add 5 + (-3)

• Find the factors of -2

You are correct if you said to multiply

(5) * (-3).

You now want to find the factors of -15. What are the factors of -15?

Multiple Select

5x² - 2x - 3

What are the factors of -15?

(Check all that apply)

• (-15)(1)

• (-1)(15)

• (5)(-3)

• (-3)(5)

Yes, they are all factors of -15. Since the product is negative, you have to have one positive and one negative factor.

Which factors add up to the coefficient of the middle term of

5x² - 2x - 3?

Multiple Choice

5x² - 2x - 3

Which factors of -15 add up to the coefficient of the middle term?

• (-1)(15)

• (-15)(1)

• (-3)(5)

• (3)(-5)

(-5)(3) = -15, and -5 + 3 = -2

Rewrite the quadratic separating the middle term into the two numbers that make up its sum.

5x² - 2x - 3 = 5x² - 5x + 3x - 3

What would be the next step to factor by grouping?

Multiple Select

5x² - 2x - 3 = 5x² - 5x + 3x - 3

What would be the next step to factor by grouping?

• 5x² (- 5x + 3x) - 3

• (5x² - 5x) + (3x - 3)

5x² - 2x - 3 = (5x² - 5x) + (3x - 3)

What can you factor out of the first group, and the second group?

Multiple Choice

5x² -2x -3 = (5x² - 5x) + (3x - 3)

What can you factor out of the first group, and what can you factor out of the second group?

• -5, 3

• 5x, 3x

• 5x, 3

• 5x, -3

5x² -2x -3 = (5x² - 5x) + (3x - 3) =

5x (x-1) + 3(x - 1)

What is the common factor in both groups?

Multiple Select

5x² -2x -3 = (5x² - 5x) + (3x - 3) =

5x (x-1) + 3(x - 1)

What is the common factor in both groups?

• 3x - 3

• 5x - 5

• x-1

• x+1

5x² -2x -3 = (5x² - 5x) + (3x - 3) =

5x (x-1) + 3(x - 1)

What is the common factor in both groups?

(x-1) is the common factor.

When you factor (x-1) from both groups, what is left?

Multiple Choice

5x² -2x -3 = (5x² - 5x) + (3x - 3) =

5x (x-1) + 3(x - 1)

When you factor (x-1) from both groups, what is left?

• (5x +3)

• (x-1)

• 5x(x-1)

• 3 (x-1)

Let's try one more

4x² -10x +6

Multiple Choice

Remember the first thing to check when factoring is to see if there is a monomial term that you can factor.

What monomial term can you factor from 4x² -10x +6?

• 4

• x

• 4x

• 2

If you answered 2, you are correct.

When you factor 2 out of 4x² -10x +6, what is the remaining expression?

Multiple Select

When you factor 2 from 4x² -10x +6, what is the remaining expression?

• 2x² - 5x + 3

• 4x² - 5x + 3

• 2x² - 5x + 6

• 2x² + 5x + 3

Now you are ready to factor by grouping.

(2)(2x² - 5x + 3).

Remember to put the 2 back as a factor at the very end, but for now, concentrate on 2x² -5x + 3

What is the first step to factor 2x² -5x + 3 by grouping?

Multiple Choice

What is the first step to factor 2x² -5x + 3 by grouping?

• Multiply -5 *3

• Multiply 2*3

• Add 2 + 3

• Look for the factors of 3

2x² -5x + 3

If you answered multiply 2 * 3, you are correct. Now you need to look for the factors of 6 that add up to the coefficient of the middle term.

Notice that the middle term is negative, but the last term is positive. What does that tell you about the factors?

Multiple Choice

2x² -5x + 3

2 * 3 is +6, but the middle term is negative. What does that tell you about the sign of your factors?

• They are both negative

• They are both positive

• One is negative and one is positive

2x² -5x + 3

If the last term is positive, and the middle term is negative, then both factors have to be negative. Remember a (-) * (-) is (+)

What are the factors of 3 that add up to the coefficient of the middle term?

Multiple Choice

2x² -5x + 3

What are the factors of 6 that add up to the coefficient of the middle term?

• (2)(3)

• (-1)(-6)

• (-2)(-3)

• (1)(6)

2x² -5x + 3

The factors you are looking for are (-2) and (-3).

How would you rewrite the quadratic using that information?

Multiple Select

2x² -5x + 3.

How would you use the factors (-2) and (-3) to rewrite the quadratic expression? (check all that apply)

• 2x² -3x - 2x + 3

• 2x² -5x + 6

• 2x² -5x + 3

• 2x² -2x -3x + 3

2x² -5x + 3

That's right. The order that you put the factors into the equation won't change the final outcome. You will just have a different common factor.

Let's use 2x² -3x -2x+ 3

Multiple Choice

2x² -3x -2x+ 3 becomes x(2x - 3) - 1(2x -3). If there are no common factors, in this case, we always factor out 1, or -1. Since the sign in front of the x was (-), -1 was factored out. Why does the sign change in front of the 3?

• I have no idea

• Because -1 * -3 = +3

2x² -5x + 3

x(2x - 3) -1(2x-3)

What does this look like when you factor out the common factor?

Multiple Choice

2x² -5x + 3

x(2x - 3) -1(2x-3)

What does this look like when you factor out the common factor?

• (2x - 3) (x+ 1)

• (2x - 3)(x-1)

• (x-3)(x-1)

• (x+3)(x+1)

That's all folks!

Or is it?

What is the final factored form of 4x² - 10x +6?

Multiple Select

What is the final factored form of 4x² - 10x +6?

• (2x-3)(x-1)

• 2(2x-3)(x-1)

Always remember if you factor out a monomial term, to include it in your final answer.

Now try a couple on your own.

Multiple Choice

Factor:

2x² - 7x + 6

• (x -1)(2x - 6)

• (x - 3) (2x - 2)

• (x - 2) (2x - 3)

• (x - 6) (2x - 1)

Multiple Choice

Factor:

x² + 16x + 15

• (x+1)(x+15)

• (x+3)(x+5)

• (x-3)(x-5)

• (x-1)(x+15)

Multiple Choice

Factor:

3v² - 4v - 7

• (3v-7)(v+1)

• 3(v-7)(v-1)

• (3v+1)(v-9)

• (3v+1)(v-10)