Section 7 – 4:
Factoring Trinomials of the form Ax 2 + Bx + C with A > 1
Selected Worked Homework Problems
Factor.
1. 2x 2 + 5x + 3!
Step 1: The GCF must be taken out first (if there is one) before factoring the hard trinomial.
There is no GCF so factor 2x 2 + 5x + 3
Step 2: Create an Easy Trinomial by moving the coefficient of the x 2 term to the end of the trinomial
and multiplying the constant term by that coefficient
Move the 2 to the right end of the trinomial and multiply it by the constant 3 to get 6
2x 2 + 5x + 3• 2 to get the easy trinomial
x 2 + 5x + 6
Step 3: Factor the easy trinomial
( x + 2) ( x + 3 )
Step 4: In step 1 you multiplied the constant –1 by the coefficient of the x 2 term.
Now divide BOTH of the constants that have been added or subtracted form the x term in
the factors by that the coefficient.
3⎞ ⎛
2⎞
⎛
⎜⎝ x + ⎟⎠ ⎜⎝ x + ⎟⎠
2
2
reduce each fraction that can be reduced (no mixed numbers)
3⎞
⎛
⎜⎝ x + ⎟⎠ ( x + 2 )
2
Step 5: "glide" the denominator of any remaining fractions (if there are any) in front of the x term
( 2x + 3)( x + 1)
You can check your answer by “ FOIL”ing the factors
( 2x + 3)( x + 1)
= 2x 2 + x + 4x + 3
= 2x 2 + 5x + 3
Factor.
Math 100 !
Section 7 – 4
HW WKD
!
©2016 Eitel
2. 3x 2 − 2x − 5!
Step 1. There is no GCF so factor 3x 2 − 2x − 5
Step 2:
3x 2 − 2x − 5 • 3 to get the easy trinomial
x 2 − 2x − 15
Step 3: Factor the easy trinomial ( x – 5 ) ( x + 3 )
Step 4: Divide each constant added or subtracted from x by 3
5⎞ ⎛
3⎞
⎛
x−
x+
⎝
3⎠ ⎝
3⎠
and then reduce each fraction that can be reduced (no mixed numbers)
5⎞
⎛
( x + 1)
x−
⎝
3⎠
Step 5: "glide" the denominator of any remaining fractions (if there are any) in front of the x term
Answer: ( 3x − 5) ( x + 1)
You can check your answer by “ FOIL”ing the factors
(3x − 5) (2x + 3)
= 6x 2 + 9x − 10x − 15
= 6x 2 − x − 15
Factor.
Math 100 !
Section 7 – 4
HW WKD
!
©2016 Eitel
3. 6x 2 + 7x − 3
!
Step 1. There is no GCF so factor 6x 2 + 7x − 3
6x 2 + 7x − 3• 6 to get the easy trinomial
Step 2:
x 2 + 7x − 18
Step 3: Factor the easy trinomial ( x – 2 ) ( x + 9 )
2⎞ ⎛
9⎞
⎛
Step 4: Divide each constant added or subtracted from x by 6 x −
x+
⎝
6⎠ ⎝
6⎠
and then reduce each fraction that can be reduced (no mixed numbers)
1⎞ ⎛
3⎞
⎛
x−
x+
⎝
3⎠ ⎝
2⎠
Step 5: "glide" the denominator of any remaining fractions (if there are any) in front of the x term
Answer:
(3x − 1) (2x + 3) !
You can check your answer by “ FOIL”ing the factors
(3x − 1) (2x + 3)
= 6x 2 + 7x − 2x + 3
= 6x 2 + 7x + 3
Math 100 !
Section 7 – 4
HW WKD
!
©2016 Eitel
Factor.
4. 4 x 2 − 5x − 6!
!
Step 1. There is no GCF so factor 4 x 2 − 5x − 6
4 x 2 − 5x − 6 • 4 to get the easy trinomial
Step 2:
x 2 − 5x − 24
Step 3: Factor the easy trinomial ( x + 3 ) ( x – 8 )
3⎞ ⎛
8⎞
⎛
Step 4: Divide each constant added or subtracted from x by 4 ⎜ x + ⎟ ⎜ x − ⎟
⎝
⎠
⎝
4
4⎠
3⎞
⎛
and then reduce each fraction that can be reduced (no mixed numbers) ⎜ x + ⎟ ( x − 2 )
⎝
4⎠
Step 5: "glide" the denominator of any remaining fractions (if there are any) in front of the x term
Answer:
( 4x + 3)( x − 2 )
!
You can check your answer by “ FOIL”ing the factors
( 4x + 3)( x − 2 )
= 4x 2 + 3x − 8x − 6
= 4x 2 − 5x + 6
!
Math 100 !
!
!
Section 7 – 4
HW WKD
!
!
!
!
©2016 Eitel
Factor.
5. 8x 2 −17x + 2!
!
Step 1. There is no GCF so factor 8x 2 −17x + 2
8x 2 − 17x + 2 •8 to get the easy trinomial
Step 2:
x 2 − 17x + 16
Step 3: Factor the easy trinomial ( x –- 1) ( x – 16 )
1⎞ ⎛
16 ⎞
⎛
Step 4: Divide each constant added or subtracted from x by 8 ⎜ x − ⎟ ⎜ x − ⎟
⎝
⎠
⎝
8
8⎠
1⎞
⎛
and then reduce each fraction that can be reduced (no mixed numbers) ⎜ x − ⎟ ( x − 2 )
⎝
8⎠
Step 5: "glide" the denominator of any remaining fractions (if there are any) in front of the x term
Answer:
( 8x − 1) ( x − 2 )
!
You can check your answer by “ FOIL”ing the factors
( 8x − 1) ( x − 2 )
= 16x 2 − 1x − 16x + 2
= 8x 2 − 16x + 2
Math 100 !
Section 7 – 4
HW WKD
!
©2016 Eitel
Factor.
6. 3x 2 + 7x + 8!
!
Step 1. There is no GCF so factor 3x 2 + 7x + 8
3x 2 + 7x + 8 • 3 to get the easy trinomial
Step 2:
x 2 + 7x + 24
Step 3: Factor the easy trinomial x 2 + 7x + 24 DOES NOT FACTOR so
3x 2 + 7x + 8 DOES NOT FACTOR
14.
9x 2 + 6x + 1
2
Step 1. There is no GCF so factor 9x + 6x + 1
9x 2 + 6x + 1• 9 to get the easy trinomial
Step 2:
x 2 + 6x + 9
Step 3: Factor the easy trinomial ( x + 3) ( x + 3)
3⎞ ⎛
3⎞
⎛
Step 4: Divide each constant added or subtracted from x by 9 x +
x+
⎝
9⎠ ⎝
9⎠
and then reduce each fraction that can be reduced (no mixed numbers)
1⎞ ⎛
1⎞
⎛
x+
x+
⎝
3⎠ ⎝
3⎠
Step 5: "glide" the denominator of any remaining fractions (if there are any) in front of the x term
Answer:
(3x + 1) ( 3x + 1)
Math 100 !
Section 7 – 4
HW WKD
!
©2016 Eitel
Factor.
15. 5x 2 − 7x + 1!
!
Step 1. There is no GCF so factor 5x 2 − 7x + 1
5x 2 − 7x + 1• 5 to get the easy trinomial
Step 2:
x 2 − 7x + 5
Step 3: Factor the easy trinomial x 2 − 7x + 5 DOES NOT FACTOR so
5x 2 − 7x + 1 DOES NOT FACTOR
Factor.
2
16. 12x + 5x − 3 !
!
2
Step 1. There is no GCF so factor 12x + 5x − 3
12x 2 + 5x − 3•12 to get the easy trinomial
Step 2:
x 2 + 5x − 36
Step 3: Factor the easy trinomial ( x – 4) ( x + 9)
4 ⎞⎛
9⎞
⎛
Step 4: Divide each constant added or subtracted from x by 12 ⎜ x − ⎟ ⎜ x + ⎟
⎝
⎠
⎝
12
12 ⎠
1⎞ ⎛
3⎞
⎛
and then reduce each fraction that can be reduced (no mixed numbers) ⎜ x − ⎟ ⎜ x + ⎟
⎝
3⎠ ⎝
4⎠
Step 5: "glide" the denominator of any remaining fractions (if there are any) in front of the x term
Answer:
( 3x − 1)( 4x + 3)
Math 100 !
Section 7 – 4
HW WKD
!
©2016 Eitel
!
Factoring Completely:
Step 1: Factor out the GCF Step 2: Factor the expression inside the parenthesis.
!
Factor.
37. 4 x 2 + 10x + 6!
!
Step 1. The GCF is 2. Factor out a 2
(
)
4 x 2 + 10x + 6= 2 2x 2 + 5x + 3 now factor 2x 2 + 5x + 3
Step 2:
2x 2 + 5x + 3• 2 to get the easy trinomial
x 2 + 5x + 6
Step 3: Factor the easy trinomial 2 ( x +3 ) ( x + 2 )
3⎞ ⎛
2⎞
⎛
Step 4: Divide each constant added or subtracted from x by 2 2 ⎜ x + ⎟ ⎜ x + ⎟
⎝
2⎠ ⎝
2⎠
2⎞
⎛
and then reduce each fraction that can be reduced (no mixed numbers) 2 ⎜ x + ⎟ ( x + 1)
⎝
3⎠
Step 5: "glide" the denominator of any remaining fractions (if there are any) in front of the x term
Answer: 2 ( 2x + 3) ( x + 1)
Math 100 !
Section 7 – 4
HW WKD
!
©2016 Eitel
!
Factoring Completely:
Step 1: Factor out the GCF Step 2: Factor the expression inside the parenthesis.
!
Factor.
38. 6x 2 + 10x − 4!
!
Step 1. The GCF is 2. Factor out a 2
(
)
6x 2 + 10x − 4 = 2 3x 2 + 5x − 2 now factor 3x 2 + 5x − 2
Step 2:
3x 2 + 5x − 2 • 3 to get the easy trinomial
x 2 + 5x − 6
Step 3: Factor the easy trinomial 2 ( x – 1 ) ( x + 6 )
1⎞ ⎛
6⎞
⎛
Step 4: Divide each constant added or subtracted from x by 3 2 ⎜ x − ⎟ ⎜ x + ⎟
⎝
3⎠ ⎝
3⎠
1⎞
⎛
and then reduce each fraction that can be reduced (no mixed numbers) 2 ⎜ x − ⎟ ( x + 2 )
⎝
3⎠
Step 5: "glide" the denominator of any remaining fractions (if there are any) in front of the x term
Answer: 2 ( 3x − 1) ( x + 2 )
Math 100 !
Section 7 – 4
HW WKD
!
©2016 Eitel
Factor Completely.
39. 6x 2 − 30x + 9!
!
Step 1. The GCF is 3. Factor out a 3
(
)
6x 2 − 30x + 9 = 3 2x 2 − 10x + 3 now factor 2x 2 − 10x + 3
Step 2:
2x 2 − 10x + 3• 2 to get the easy trinomial
x 2 − 10x + 6
Step 3: Factor the easy trinomial x 2 − 10x + 6 DOES NOT FACTOR so
Answer: 3 ( 2x 2 − 10x + 3)
Note: DNF is not correct. You factored out a 3 so the original polynomial DID factor
but 3 ( 2x 2 − 10x + 3) does not factor any further,
Math 100 !
Section 7 – 4
HW WKD
!
©2016 Eitel