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Algebra 1

Factor Trinomials: Mini-Lesson Review

Algebra 1Factor Trinomials: Mini-Lesson Review
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Mini Lesson Question

Factor: y 2 − y − 42 .
  1. ( y − 7 ) ( y − 6 )
  2. ( y + 7 ) ( y + 6 )
  3. ( y + 7 ) ( y − 6 )
  4. ( y − 7 ) ( y + 6 )

Factor Trinomials

Example 1

To factor the trinomial x 2 + 8 x + 12 x 2 + 8 x + 12 , you can reverse the process of FOIL.

The trinomial factors to two binomials of the form ( x + m ) ( x + n ) ( x + m ) ( x + n ) .

  • Using FOIL: ( x + m ) ( x + n ) = x 2 + n x + m x + m n ( x + m ) ( x + n ) = x 2 + n x + m x + m n .
  • Using the distributive property: x 2 + ( n + m ) x + m n x 2 + ( n + m ) x + m n .

Take a look at the trinomial x 2 + 8 x + 12 x 2 + 8 x + 12 :

Image showing x squared plus the quantity of n plus m times x plus m times n. On the next line, the trinomial is rewritten as x squared plus 8 times x plus 12 and arrows point from the quantity of n plus m in the trinomial above to the 8, and an arrow from m times n in the trinomial above to 12.

To factor the trinomial, we need factors of 12, m m and n n , so that the sum, n + m n + m , is 8.

Use a table to find all the different combinations of factors of 12. Then, find the sum of the factors.

Factors of 12 Sum of factors

1 , 12 1 , 12

1 + 12 = 13 1 + 12 = 13

2 , 6 2 , 6

2 + 6 = 8 ✓ 2 + 6 = 8 ✓

3 , 4 3 , 4

3 + 4 = 7 3 + 4 = 7

The numbers 2 and 6 have a product of 12 and a sum of 8. They are the factors we need:

x 2 + 8 x + 12 = ( x + 6 ) ( x + 2 ) x 2 + 8 x + 12 = ( x + 6 ) ( x + 2 )

Example 2

Factor the trinomial x 2 − 5 x − 24 x 2 − 5 x − 24 .

To factor the trinomial, we need factors of -24, m m and n n , so that the sum, n + m n + m is -5.

  • The product of the factors is negative, so the factors must have opposite signs. 
  • The sum is negative, so the negative sign will go with the factor with the greater absolute value.
Factors of 12 Sum of factors

1 , − 24 1 , − 24

2 , − 12 2 , − 12

3 , − 8 3 , − 8

4 , − 6 4 , − 6

1 + ( − 24 ) = − 23 1 + ( − 24 ) = − 23

2 + ( − 12 ) = − 10 2 + ( − 12 ) = − 10

3 + ( − 8 ) = − 5 ✓ 3 + ( − 8 ) = − 5 ✓

4 + ( − 6 ) = − 2 4 + ( − 6 ) = − 2

The numbers 3 and -8 have a product of -24 and a sum of -5.

x 2 − 5 x − 24 = ( x + 3 ) ( x − 8 ) x 2 − 5 x − 24 = ( x + 3 ) ( x − 8 )

Example 3

Factor the trinomial x 2 − 9 x + 18 x 2 − 9 x + 18 .

To factor the trinomial, we need factors of 18, m m and n n , so that the sum, n + m n + m , is -9.

  • The product of the factors is positive, so the factors must have the same signs. 
  • The sum is negative, so both factors are negative.
Factors of 18 Sum of factors

− 1 , − 18 − 1 , − 18

− 2 , − 9 − 2 , − 9

− 3 , − 6 − 3 , − 6

− 1 + ( − 18 ) = − 19 − 1 + ( − 18 ) = − 19

− 2 + ( − 9 ) = − 11 − 2 + ( − 9 ) = − 11

− 3 + ( − 6 ) = − 9 ✓ − 3 + ( − 6 ) = − 9 ✓

The numbers -3 and -6 have a product of 18 and a sum of -9.

x 2 − 9 x + 18 = ( x − 3 ) ( x − 6 ) x 2 − 9 x + 18 = ( x − 3 ) ( x − 6 )

Try it

Factoring Trinomials

Factor the trinomial x 2 + 2 x − 24 x 2 + 2 x − 24 .

Check Your Understanding

Factor: y 2 − 13 y + 36 y 2 − 13 y + 36 .

  1. ( y − 4 ) ( y − 9 ) ( y − 4 ) ( y − 9 )

  2. ( y − 2 ) ( y + 18 ) ( y − 2 ) ( y + 18 )

  3. ( y + 4 ) ( y + 9 ) ( y + 4 ) ( y + 9 )

  4. ( y − 6 ) ( y − 6 ) ( y − 6 ) ( y − 6 )

Video: Factoring Trinomials

Watch the following video to learn more about factoring trinomials.

Khan Academy: Factoring Quadratics

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