Algebra 1 · Unit 7
Factor ax² + bx + c with a > 1 using Slide and Divide. Take out a GCF first when you can.
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Slide and Divide
When a > 1, slide a onto the constant, factor the new a = 1 trinomial, then divide the constants in the factors by a and reduce.
GCF first
If the three terms share a GCF, factor it out so the leftover leading coefficient has few factors.
Check
Multiply the binomials (and the GCF). You should get the original polynomial.
Grouping is the rewrite
After you find the pair, the four-term grouping is the same work in a different order.
a = 2 or 3. Positive terms. Slide and Divide.
2x² + 13x + 6
SLIDE, multiply 2*6
2x² + 13x + 6
Solve the Puzzle. What two numbers multiply to 12 and add to 13.
1*12
2*6
3*4
2x²
+ 6
Now we divided 13x into two pieces 1x and 12x.
2x² + 1x + 12x + 6
Factor by grouping.
2x² + 1x + 12x + 6
Factor the GCF from each part.
x(2x+1) + 6(2x+1)
(2x+1)(x+6)
Answer: (2x+1)(x+6)
2x² + 15x + 7
SLIDE, multiply 2*7
2x² + 15x + 7
Solve the Puzzle. What two numbers multiply to 14 and add to 15.
1*14
2*7
2x²
+ 7
Now we divided 15x into two pieces 1x and 14x.
2x² + 1x + 14x + 7
Factor by grouping.
2x² + 1x + 14x + 7
Factor the GCF from each part.
x(2x+1) + 7(2x+1)
(2x+1)(x+7)
Answer: (2x+1)(x+7)
3x² + 14x + 8
SLIDE, multiply 3*8
3x² + 14x + 8
Solve the Puzzle. What two numbers multiply to 24 and add to 14.
1*24
2*12
3*8
4*6
3x²
+ 8
Now we divided 14x into two pieces 2x and 12x.
3x² + 2x + 12x + 8
Factor by grouping.
3x² + 2x + 12x + 8
Factor the GCF from each part.
x(3x+2) + 4(3x+2)
(3x+2)(x+4)
Answer: (3x+2)(x+4)
2x² + 17x + 8
SLIDE, multiply 2*8
2x² + 17x + 8
Solve the Puzzle. What two numbers multiply to 16 and add to 17.
1*16
2*8
4*4
2x²
+ 8
Now we divided 17x into two pieces 1x and 16x.
2x² + 1x + 16x + 8
Factor by grouping.
2x² + 1x + 16x + 8
Factor the GCF from each part.
x(2x+1) + 8(2x+1)
(2x+1)(x+8)
Answer: (2x+1)(x+8)
3x² + 16x + 5
SLIDE, multiply 3*5
3x² + 16x + 5
Solve the Puzzle. What two numbers multiply to 15 and add to 16.
1*15
3*5
3x²
+ 5
Now we divided 16x into two pieces 1x and 15x.
3x² + 1x + 15x + 5
Factor by grouping.
3x² + 1x + 15x + 5
Factor the GCF from each part.
x(3x+1) + 5(3x+1)
(3x+1)(x+5)
Answer: (3x+1)(x+5)
2x² + 19x + 9
SLIDE, multiply 2*9
2x² + 19x + 9
Solve the Puzzle. What two numbers multiply to 18 and add to 19.
1*18
2*9
3*6
2x²
+ 9
Now we divided 19x into two pieces 1x and 18x.
2x² + 1x + 18x + 9
Factor by grouping.
2x² + 1x + 18x + 9
Factor the GCF from each part.
x(2x+1) + 9(2x+1)
(2x+1)(x+9)
Answer: (2x+1)(x+9)
A negative term. Slide still works.
2x² + 3x − 9
SLIDE, multiply 2*(−9)
2x² + 3x − 9
Solve the Puzzle. What two numbers multiply to −18 and add to 3.
(−1)*18
1*(−18)
(−2)*9
2*(−9)
(−3)*6
3*(−6)
2x²
− 9
Now we divided 3x into two pieces −3x and 6x.
2x² − 3x + 6x − 9
Factor by grouping.
2x² − 3x + 6x − 9
Factor the GCF from each part.
x(2x−3) + 3(2x−3)
(2x−3)(x+3)
Answer: (2x−3)(x+3)
2x² − 9x + 4
SLIDE, multiply 2*4
2x² − 9x + 4
Solve the Puzzle. What two numbers multiply to 8 and add to −9.
(−1)*(−8)
(−2)*(−4)
2x²
+ 4
Now we divided −9x into two pieces −1x and −8x.
2x² − 1x − 8x + 4
Factor by grouping.
2x² − 1x − 8x + 4
Factor the GCF from each part.
x(2x−1) − 4(2x−1)
(2x−1)(x−4)
Answer: (2x−1)(x−4)
3x² + 4x − 4
SLIDE, multiply 3*(−4)
3x² + 4x − 4
Solve the Puzzle. What two numbers multiply to −12 and add to 4.
(−1)*12
1*(−12)
(−2)*6
2*(−6)
(−3)*4
3*(−4)
3x²
− 4
Now we divided 4x into two pieces −2x and 6x.
3x² − 2x + 6x − 4
Factor by grouping.
3x² − 2x + 6x − 4
Factor the GCF from each part.
x(3x−2) + 2(3x−2)
(3x−2)(x+2)
Answer: (3x−2)(x+2)
2x² − 3x − 20
SLIDE, multiply 2*(−20)
2x² − 3x − 20
Solve the Puzzle. What two numbers multiply to −40 and add to −3.
(−1)*40
1*(−40)
(−2)*20
2*(−20)
(−4)*10
4*(−10)
(−5)*8
5*(−8)
2x²
− 20
Now we divided −3x into two pieces 5x and −8x.
2x² + 5x − 8x − 20
Factor by grouping.
2x² + 5x − 8x − 20
Factor the GCF from each part.
x(2x+5) − 4(2x+5)
(2x+5)(x−4)
Answer: (2x+5)(x−4)
3x² − 7x − 6
SLIDE, multiply 3*(−6)
3x² − 7x − 6
Solve the Puzzle. What two numbers multiply to −18 and add to −7.
(−1)*18
1*(−18)
(−2)*9
2*(−9)
(−3)*6
3*(−6)
3x²
− 6
Now we divided −7x into two pieces 2x and −9x.
3x² + 2x − 9x − 6
Factor by grouping.
3x² + 2x − 9x − 6
Factor the GCF from each part.
x(3x+2) − 3(3x+2)
(3x+2)(x−3)
Answer: (3x+2)(x−3)
2x² + 5x − 12
SLIDE, multiply 2*(−12)
2x² + 5x − 12
Solve the Puzzle. What two numbers multiply to −24 and add to 5.
(−1)*24
1*(−24)
(−2)*12
2*(−12)
(−3)*8
3*(−8)
(−4)*6
4*(−6)
2x²
− 12
Now we divided 5x into two pieces −3x and 8x.
2x² − 3x + 8x − 12
Factor by grouping.
2x² − 3x + 8x − 12
Factor the GCF from each part.
x(2x−3) + 4(2x−3)
(2x−3)(x+4)
Answer: (2x−3)(x+4)
Larger a, or a GCF first.
4x² + 11x + 6
SLIDE, multiply 4*6
4x² + 11x + 6
Solve the Puzzle. What two numbers multiply to 24 and add to 11.
1*24
2*12
3*8
4*6
4x²
+ 6
Now we divided 11x into two pieces 3x and 8x.
4x² + 3x + 8x + 6
Factor by grouping.
4x² + 3x + 8x + 6
Factor the GCF from each part.
x(4x+3) + 2(4x+3)
(4x+3)(x+2)
Answer: (4x+3)(x+2)
6x² + 17x + 12
SLIDE, multiply 6*12
6x² + 17x + 12
Solve the Puzzle. What two numbers multiply to 72 and add to 17.
1*72
2*36
3*24
4*18
6*12
8*9
6x²
+ 12
Now we divided 17x into two pieces 8x and 9x.
6x² + 8x + 9x + 12
Factor by grouping.
6x² + 8x + 9x + 12
Factor the GCF from each part.
2x(3x+4) + 3(3x+4)
(3x+4)(2x+3)
Answer: (3x+4)(2x+3)
4x² − 13x + 3
SLIDE, multiply 4*3
4x² − 13x + 3
Solve the Puzzle. What two numbers multiply to 12 and add to −13.
(−1)*(−12)
(−2)*(−6)
(−3)*(−4)
4x²
+ 3
Now we divided −13x into two pieces −1x and −12x.
4x² − 1x − 12x + 3
Factor by grouping.
4x² − 1x − 12x + 3
Factor the GCF from each part.
x(4x−1) − 3(4x−1)
(4x−1)(x−3)
Answer: (4x−1)(x−3)
GCF is 2.
2(2x² + 11x + 5)
2x² + 11x + 5
SLIDE, multiply 2*5
2x² + 11x + 5
Solve the Puzzle. What two numbers multiply to 10 and add to 11.
1*10
2*5
2x²
+ 5
Now we divided 11x into two pieces 1x and 10x.
2x² + 1x + 10x + 5
Factor by grouping.
2x² + 1x + 10x + 5
Factor the GCF from each part.
x(2x+1) + 5(2x+1)
(2x+1)(x+5)
Put the 2 back.
2(2x+1)(x+5)
Answer: 2(2x+1)(x+5)
GCF is 2.
2(3x² + 10x + 3)
3x² + 10x + 3
SLIDE, multiply 3*3
3x² + 10x + 3
Solve the Puzzle. What two numbers multiply to 9 and add to 10.
1*9
3*3
3x²
+ 3
Now we divided 10x into two pieces 1x and 9x.
3x² + 1x + 9x + 3
Factor by grouping.
3x² + 1x + 9x + 3
Factor the GCF from each part.
x(3x+1) + 3(3x+1)
(3x+1)(x+3)
Put the 2 back.
2(3x+1)(x+3)
Answer: 2(3x+1)(x+3)
6x² + 7x − 3
SLIDE, multiply 6*(−3)
6x² + 7x − 3
Solve the Puzzle. What two numbers multiply to −18 and add to 7.
(−1)*18
1*(−18)
(−2)*9
2*(−9)
(−3)*6
3*(−6)
6x²
− 3
Now we divided 7x into two pieces −2x and 9x.
6x² − 2x + 9x − 3
Factor by grouping.
6x² − 2x + 9x − 3
Factor the GCF from each part.
2x(3x−1) + 3(3x−1)
(3x−1)(2x+3)
Answer: (3x−1)(2x+3)
Slide the 2 onto the 3: think x² + 7x + 6.
1 and 6 multiply to 6 and add to 7.
(x + 1)(x + 6)
Divide 1 and 6 by 2: (x + 12)(x + 3).
Slide the 2 onto the first x: (2x + 1)(x + 3).
Check: 2x(x) = 2x², 2x(3) = 6x, 1(x) = x, 1(3) = 3.
2x² + 7x + 3
Answer: (2x + 1)(x + 3)
Slide 4 onto 3: x² + 8x + 12.
2 and 6 multiply to 12 and add to 8.
(x + 2)(x + 6)
Divide 2 and 6 by 4: (x + 24)(x + 64).
Reduce: (x + 12)(x + 32).
Slide 2 onto the first x and 2 onto the second x: (2x + 1)(2x + 3).
If you skip reducing, the 2s do not land on the x’s correctly.
Answer: (2x + 1)(2x + 3)
(6x + 6)(x + 1) = 6x² + 6x + 6x + 6 = 6x² + 12x + 6, not 13x.
Slide 6 onto 6: x² + 13x + 36.
4 and 9 multiply to 36 and add to 13.
(x + 4)(x + 9)
Divide by 6 and reduce: (x + 23)(x + 32).
Slide: (3x + 2)(2x + 3).
Check: 3x(2x) = 6x², 3x(3) = 9x, 2(2x) = 4x, 2(3) = 6 → 6x² + 13x + 6.
Answer: (3x + 2)(2x + 3)
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