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Trinomial Factoring - Sample Math Practice Problems

The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. In the main program, all problems are automatically graded and the difficulty adapts dynamically based on performance. Answers to these sample questions appear at the bottom of the page. This page does not grade your responses.

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Complexity=1, Mode=simple

Factor the expression as completely as possible.
1.   x 2 - 5x + 6
2.   x 2 - 1

Complexity=3, Mode=prefix

Factor the expression as completely as possible.
1.   8x 3 + 168x 2 - 176x
2.   x 2 - 36

Complexity=4, Mode=frontConst

Factor the expression as completely as possible.
1.   -x 3 - 47x 2 - 90x
2.   - 8x 3 - 452x 2 - 224x

Complexity=5, Mode=includeY

Factor the expression as completely as possible.
1.   2x 3 + 147x 2y - 74xy 2
2.   6x 3 - 100x 2y - 34xy 2

Complexity=6, Mode=hardest

Factor the expression as completely as possible.
1.   - 24w 3x 6 - 92w 2x 4y + 220wx 2y 2
2.   5w 3x 3z 3 + 280w 2x 2yz 2 + 540wxy 2z

Answers

Complexity=1, Mode=simple

Factor the expression as completely as possible.
#ProblemCorrect AnswerYour Answer
1x 2 - 5x + 6
Solution
Front coefficient is 1. Factors are: 1
Back coefficient is 6. Factors are: 1, 2, 3, 6
Now test combinations of these factors to get the middle value of - 5x.

For each test, do a partial FOIL. Multiply the outer factors and inner factors and add them together to see if they are equal to - 5x.
Test: (x - 1)(x - 6) => - 6x + -x = - 7x
Test: (x - 2)(x - 3) => - 3x + - 2x = - 5x This combination matches.
Result: x 2 + - 5x + 6

#ProblemCorrect AnswerYour Answer
2x 2 - 1
Solution
Use formula a2 - b2 = (a+b)(a-b)
x 2 - 1 = (x + 1)(x - 1)
Result: x 2 + - 1

Complexity=3, Mode=prefix

Factor the expression as completely as possible.
#ProblemCorrect AnswerYour Answer
18x 3 + 168x 2 - 176x
Solution
Factor out 8x from 8x 3 + 168x 2 + - 176x
8x(x 2 + 21x - 22)
Front coefficient is 1. Factors are: 1
Back coefficient is -22. Factors are: 1, 2, 11, 22
Now test combinations of these factors to get the middle value of 21x.

For each test, do a partial FOIL. Multiply the outer factors and inner factors and add them together to see if they are equal to 21x.
Test: (x + 1)(x - 22) => - 22x + x = - 21x
Test: (x + 2)(x - 11) => - 11x + 2x = - 9x
Test: (x + 11)(x - 2) => - 2x + 11x = 9x
Test: (x + 22)(x - 1) => -x + 22x = 21x This combination matches.
Result: 8x 3 + 168x 2 + - 176x

#ProblemCorrect AnswerYour Answer
2x 2 - 36
Solution
Use formula a2 - b2 = (a+b)(a-b)
x 2 - 36 = (x + 6)(x - 6)
Result: x 2 + - 36

Complexity=4, Mode=frontConst

Factor the expression as completely as possible.
#ProblemCorrect AnswerYour Answer
1-x 3 - 47x 2 - 90x
Solution
Factor out -x from -x 3 + - 47x 2 + - 90x
-x(x 2 + 47x + 90)
Front coefficient is 1. Factors are: 1
Back coefficient is 90. Factors are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 45, 90
Now test combinations of these factors to get the middle value of 47x.

For each test, do a partial FOIL. Multiply the outer factors and inner factors and add them together to see if they are equal to 47x.
Test: (x + 1)(x + 90) => 90x + x = 91x
Test: (x + 2)(x + 45) => 45x + 2x = 47x This combination matches.
Result: -x 3 + - 47x 2 + - 90x

#ProblemCorrect AnswerYour Answer
2- 8x 3 - 452x 2 - 224x
Solution
Factor out - 4x from - 8x 3 + - 452x 2 + - 224x
- 4x(2x 2 + 113x + 56)
Front coefficient is 2. Factors are: 1, 2
Back coefficient is 56. Factors are: 1, 2, 4, 7, 8, 14, 28, 56
Now test combinations of these factors to get the middle value of 113x.

For each test, do a partial FOIL. Multiply the outer factors and inner factors and add them together to see if they are equal to 113x.
Test: (x + 1)(2x + 56) => 56x + 2x = 58x
Test: (x + 2)(2x + 28) => 28x + 4x = 32x
Test: (x + 4)(2x + 14) => 14x + 8x = 22x
Test: (x + 7)(2x + 8) => 8x + 14x = 22x
Test: (x + 8)(2x + 7) => 7x + 16x = 23x
Test: (x + 14)(2x + 4) => 4x + 28x = 32x
Test: (x + 28)(2x + 2) => 2x + 56x = 58x
Test: (x + 56)(2x + 1) => x + 112x = 113x This combination matches.
Result: - 8x 3 + - 452x 2 + - 224x

Complexity=5, Mode=includeY

Factor the expression as completely as possible.
#ProblemCorrect AnswerYour Answer
12x 3 + 147x 2y - 74xy 2
Solution
Factor out x from 2x 3 + 147x 2y + - 74xy 2
x(2x 2 + 147xy - 74y 2)
Front coefficient is 2. Factors are: 1, 2
Back coefficient is -74. Factors are: 1, 2, 37, 74
Now test combinations of these factors to get the middle value of 147xy.

For each test, do a partial FOIL. Multiply the outer factors and inner factors and add them together to see if they are equal to 147xy.
Test: (x + y)(2x - 74y) => - 74xy + 2xy = - 72xy
Test: (x + 2y)(2x - 37y) => - 37xy + 4xy = - 33xy
Test: (x + 37y)(2x - 2y) => - 2xy + 74xy = 72xy
Test: (x + 74y)(2x - y) => -xy + 148xy = 147xy This combination matches.
Result: 2x 3 + 147x 2y + - 74xy 2

#ProblemCorrect AnswerYour Answer
26x 3 - 100x 2y - 34xy 2
Solution
Factor out 2x from 6x 3 + - 100x 2y + - 34xy 2
2x(3x 2 - 50xy - 17y 2)
Front coefficient is 3. Factors are: 1, 3
Back coefficient is -17. Factors are: 1, 17
Now test combinations of these factors to get the middle value of - 50xy.

For each test, do a partial FOIL. Multiply the outer factors and inner factors and add them together to see if they are equal to - 50xy.
Test: (x + y)(3x - 17y) => - 17xy + 3xy = - 14xy
Test: (x + 17y)(3x - y) => -xy + 51xy = 50xy
Test: (x - y)(3x + 17y) => 17xy + - 3xy = 14xy
Test: (x - 17y)(3x + y) => xy + - 51xy = - 50xy This combination matches.
Result: 6x 3 + - 100x 2y + - 34xy 2

Complexity=6, Mode=hardest

Factor the expression as completely as possible.
#ProblemCorrect AnswerYour Answer
1- 24w 3x 6 - 92w 2x 4y + 220wx 2y 2
Solution
Factor out - 4wx 2 from - 24w 3x 6 + - 92w 2x 4y + 220wx 2y 2
- 4wx 2(6w 2x 4 + 23wx 2y - 55y 2)
Front coefficient is 6. Factors are: 1, 2, 3, 6
Back coefficient is -55. Factors are: 1, 5, 11, 55
Now test combinations of these factors to get the middle value of 23wx 2y.

For each test, do a partial FOIL. Multiply the outer factors and inner factors and add them together to see if they are equal to 23wx 2y.
Test: (wx 2 + y)(6wx 2 - 55y) => - 55wx 2y + 6wx 2y = - 49wx 2y
Test: (wx 2 + 5y)(6wx 2 - 11y) => - 11wx 2y + 30wx 2y = 19wx 2y
Test: (wx 2 + 11y)(6wx 2 - 5y) => - 5wx 2y + 66wx 2y = 61wx 2y
Test: (wx 2 + 55y)(6wx 2 - y) => -wx 2y + 330wx 2y = 329wx 2y
Test: (wx 2 - y)(6wx 2 + 55y) => 55wx 2y + - 6wx 2y = 49wx 2y
Test: (wx 2 - 5y)(6wx 2 + 11y) => 11wx 2y + - 30wx 2y = - 19wx 2y
Test: (wx 2 - 11y)(6wx 2 + 5y) => 5wx 2y + - 66wx 2y = - 61wx 2y
Test: (wx 2 - 55y)(6wx 2 + y) => wx 2y + - 330wx 2y = - 329wx 2y
Test: (2wx 2 + y)(3wx 2 - 55y) => - 110wx 2y + 3wx 2y = - 107wx 2y
Test: (2wx 2 + 5y)(3wx 2 - 11y) => - 22wx 2y + 15wx 2y = - 7wx 2y
Test: (2wx 2 + 11y)(3wx 2 - 5y) => - 10wx 2y + 33wx 2y = 23wx 2y This combination matches.
Result: - 24w 3x 6 + - 92w 2x 4y + 220wx 2y 2

#ProblemCorrect AnswerYour Answer
25w 3x 3z 3 + 280w 2x 2yz 2 + 540wxy 2z
Solution
Factor out 5wxz from 5w 3x 3z 3 + 280w 2x 2yz 2 + 540wxy 2z
5wxz(w 2x 2z 2 + 56wxyz + 108y 2)
Front coefficient is 1. Factors are: 1
Back coefficient is 108. Factors are: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108
Now test combinations of these factors to get the middle value of 56wxyz.

For each test, do a partial FOIL. Multiply the outer factors and inner factors and add them together to see if they are equal to 56wxyz.
Test: (wxz + y)(wxz + 108y) => 108wxyz + wxyz = 109wxyz
Test: (wxz + 2y)(wxz + 54y) => 54wxyz + 2wxyz = 56wxyz This combination matches.
Result: 5w 3x 3z 3 + 280w 2x 2yz 2 + 540wxy 2z

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