H Algebra 2/Trig · Unit 0
Group the terms in pairs. Factor each pair, then pull out the matching binomial.
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Factor by grouping
1. Check all four terms for a GCF. If there is one, factor it out first. 2. Split into two pairs. Put a plus between the groups. 3. Factor the GCF out of each pair. 4. The two groups should share a matching binomial. Factor that binomial out front. 5. Write the leftover pieces as the second binomial. Check if anything factors further.
Worked example
x³ + 5x² + 2x + 10
No GCF for all four terms.
x³ + 5x² + 2x + 10
Factor by grouping
x²(x+5) + 2(x+5)
(x+5)(x²+2)
Factor. x³ + 5x² + 2x + 10
No GCF for all four terms.
x³ + 5x² + 2x + 10
Factor by grouping
x²(x+5) + 2(x+5)
(x+5)(x²+2)
Answer: (x+5)(x²+2)
Factor. x³ + 3x² + 4x + 12
No GCF for all four terms.
x³ + 3x² + 4x + 12
Factor by grouping
x²(x+3) + 4(x+3)
(x+3)(x²+4)
Answer: (x+3)(x²+4)
Factor. x³ + 2x² + 5x + 10
No GCF for all four terms.
x³ + 2x² + 5x + 10
Factor by grouping
x²(x+2) + 5(x+2)
(x+2)(x²+5)
Answer: (x+2)(x²+5)
Factor. x³ + 4x² + x + 4
No GCF for all four terms.
x³ + 4x² + x + 4
Factor by grouping
x²(x+4) + 1(x+4)
(x+4)(x²+1)
Answer: (x+4)(x²+1)
Factor. 2x³ + 6x² + x + 3
No GCF for all four terms.
2x³ + 6x² + x + 3
Factor by grouping
2x²(x+3) + 1(x+3)
(x+3)(2x²+1)
Answer: (x+3)(2x²+1)
Factor. 3x³ + 6x² + 2x + 4
No GCF for all four terms.
3x³ + 6x² + 2x + 4
Factor by grouping
3x²(x+2) + 2(x+2)
(x+2)(3x²+2)
Answer: (x+2)(3x²+2)
Factor. x³ + 4x² − 2x − 8
No GCF for all four terms.
x³ + 4x² − 2x + 8
Factor by grouping
x²(x+4) − 2(x+4)
(x+4)(x²−2)
Answer: (x+4)(x²−2)
Factor. x³ − 3x² + 5x − 15
No GCF for all four terms.
x³ − 3x² + 5x − 15
Factor by grouping
x²(x−3) + 5(x−3)
(x−3)(x²+5)
Answer: (x−3)(x²+5)
Caution! Look for Difference of Squares.
Factor. x³ + 7x² − 4x − 28
No GCF for all four terms.
x³ + 7x² − 4x + 28
Factor by grouping
x²(x+7) − 4(x+7)
(x+7)(x²−4)
x² − 4 is a difference of squares.
(x−2)(x+2)
(x+7)(x−2)(x+2)
Answer: (x+7)(x−2)(x+2)
Factor. x³ + 3x² − 4x − 12
No GCF for all four terms.
x³ + 3x² − 4x + 12
Factor by grouping
x²(x+3) − 4(x+3)
(x+3)(x²−4)
x² − 4 is a difference of squares.
(x−2)(x+2)
(x+3)(x−2)(x+2)
Answer: (x+3)(x−2)(x+2)
Factor. x³ − 2x² − 9x + 18
No GCF for all four terms.
x³ − 2x² − 9x − 18
Factor by grouping
x²(x−2) − 9(x−2)
(x−2)(x²−9)
x² − 9 is a difference of squares.
(x−3)(x+3)
(x−2)(x−3)(x+3)
Answer: (x−2)(x−3)(x+3)
Factor. x³ + 2x² − 16x − 32
No GCF for all four terms.
x³ + 2x² − 16x + 32
Factor by grouping
x²(x+2) − 16(x+2)
(x+2)(x²−16)
x² − 16 is a difference of squares.
(x−4)(x+4)
(x+2)(x−4)(x+4)
Answer: (x+2)(x−4)(x+4)
Factor. x³ − 5x² − 4x + 20
No GCF for all four terms.
x³ − 5x² − 4x − 20
Factor by grouping
x²(x−5) − 4(x−5)
(x−5)(x²−4)
x² − 4 is a difference of squares.
(x−2)(x+2)
(x−5)(x−2)(x+2)
Answer: (x−5)(x−2)(x+2)
Factor. x³ + 6x² − 9x − 54
No GCF for all four terms.
x³ + 6x² − 9x + 54
Factor by grouping
x²(x+6) − 9(x+6)
(x+6)(x²−9)
x² − 9 is a difference of squares.
(x−3)(x+3)
(x+6)(x−3)(x+3)
Answer: (x+6)(x−3)(x+3)
Factor. 2x³ + 8x² + 6x + 24
GCF of all four terms is 2: 2(x³ + 4x² + 3x + 12).
x³ + 4x² + 3x + 12
Factor by grouping
x²(x+4) + 3(x+4)
2(x+4)(x²+3)
Answer: 2(x+4)(x²+3)
Factor. 3x³ + 9x² + 6x + 18
GCF of all four terms is 3: 3(x³ + 3x² + 2x + 6).
x³ + 3x² + 2x + 6
Factor by grouping
x²(x+3) + 2(x+3)
3(x+3)(x²+2)
Answer: 3(x+3)(x²+2)
Factor. 2x³ − 2x² − 8x + 8
GCF of all four terms is 2: 2(x³ − x² − 4x + 4).
x³ − x² − 4x − 4
Factor by grouping
x²(x−1) − 4(x−1)
2(x−1)(x²−4)
x² − 4 is a difference of squares.
(x−2)(x+2)
2(x−1)(x−2)(x+2)
Answer: 2(x−1)(x−2)(x+2)
Factor. 4x³ + 4x² − 12x − 12
GCF of all four terms is 4: 4(x³ + x² − 3x − 3).
x³ + x² − 3x + 3
Factor by grouping
x²(x+1) − 3(x+1)
4(x+1)(x²−3)
Answer: 4(x+1)(x²−3)
Group the x-only terms, then the terms with a y. Factor by grouping.
Factor. 2x³ + 8x²y + 12xy + 3x² − 5x − 20y
Group the x-only terms.
2x³ + 3x² − 5x
Group the terms with a y.
8x²y + 12xy − 20y
Factor by grouping.
Factor x from the x-only terms.
Factor 4y from the y terms.
x(2x² + 3x − 5) + 4y(2x² + 3x − 5)
(2x² + 3x − 5)(x+4y)
Factor 2x² + 3x − 5.
(2x+5)(x−1)
(2x+5)(x−1)(x+4y)
Answer: (2x+5)(x−1)(x+4y)
Factor. x³ + 2x²y + 10xy + 5x² + 6x + 12y
Group the x-only terms.
x³ + 5x² + 6x
Group the terms with a y.
2x²y + 10xy + 12y
Factor by grouping.
Factor x from the x-only terms.
Factor 2y from the y terms.
x(x² + 5x + 6) + 2y(x² + 5x + 6)
(x² + 5x + 6)(x+2y)
Factor x² + 5x + 6.
(x+2)(x+3)
(x+2)(x+3)(x+2y)
Answer: (x+2)(x+3)(x+2y)
Factor. 2x³ + 6x²y + 3xy + x² − 6x − 18y
Group the x-only terms.
2x³ + x² − 6x
Group the terms with a y.
6x²y + 3xy − 18y
Factor by grouping.
Factor x from the x-only terms.
Factor 3y from the y terms.
x(2x² + x − 6) + 3y(2x² + x − 6)
(2x² + x − 6)(x+3y)
Factor 2x² + x − 6.
(2x−3)(x+2)
(2x−3)(x+2)(x+3y)
Answer: (2x−3)(x+2)(x+3y)
Factor. x³ − 2x²y + 8xy − 4x² − 5x + 10y
Group the x-only terms.
x³ − 4x² − 5x
Group the terms with a y.
−2x²y + 8xy + 10y
Factor by grouping.
Factor x from the x-only terms.
Factor −2y from the y terms.
x(x² − 4x − 5) − 2y(x² − 4x − 5)
(x² − 4x − 5)(x−2y)
Factor x² − 4x − 5.
(x−5)(x+1)
(x−5)(x+1)(x−2y)
Answer: (x−5)(x+1)(x−2y)
Factor. 2x³ + 8x²y − 28xy − 7x² + 3x + 12y
Group the x-only terms.
2x³ − 7x² + 3x
Group the terms with a y.
8x²y − 28xy + 12y
Factor by grouping.
Factor x from the x-only terms.
Factor 4y from the y terms.
x(2x² − 7x + 3) + 4y(2x² − 7x + 3)
(2x² − 7x + 3)(x+4y)
Factor 2x² − 7x + 3.
(2x−1)(x−3)
(2x−1)(x−3)(x+4y)
Answer: (2x−1)(x−3)(x+4y)
Factor. x³ − 3x²y − 6xy + 2x² − 15x + 45y
Group the x-only terms.
x³ + 2x² − 15x
Group the terms with a y.
−3x²y − 6xy + 45y
Factor by grouping.
Factor x from the x-only terms.
Factor −3y from the y terms.
x(x² + 2x − 15) − 3y(x² + 2x − 15)
(x² + 2x − 15)(x−3y)
Factor x² + 2x − 15.
(x+5)(x−3)
(x+5)(x−3)(x−3y)
Answer: (x+5)(x−3)(x−3y)
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