Question

topic 5: factoring expressions
directions: factor each expression. if it cannot be factored, write \prime.\

1. ( 14x - 20 )
2. ( 32y + 56 )
3. ( 9x - 7 )
4. ( 3x + 9y )
5. ( 8m + 35n )
6. ( 15x + 60y )
7. ( 5 + 2(-6y - 2) - 5y )
8. ( 9 - 4(2p + 3) - 8p + 7 )
9. ( -1x - (x - 3) - 12 + 4x )
10. ( -x + z(3x + y) - 5y + 10x )

© gina wilson (all things algebra®, llc), 2017

Explanation:

Step1: Factor \(14x - 20\)

Find the greatest common factor (GCF) of \(14x\) and \(20\). The GCF of \(14\) and \(20\) is \(2\).
\(14x - 20 = 2(7x - 10)\)

Step2: Factor \(32y + 56\)

Find the GCF of \(32y\) and \(56\). The GCF of \(32\) and \(56\) is \(8\).
\(32y + 56 = 8(4y + 7)\)

Step3: Analyze \(9x - 7\)

The GCF of \(9x\) and \(7\) is \(1\), so it is prime.

Step4: Factor \(3x + 9y\)

Find the GCF of \(3x\) and \(9y\). The GCF of \(3\) and \(9\) is \(3\).
\(3x + 9y = 3(x + 3y)\)

Step5: Factor \(8m + 35n\)

The GCF of \(8m\) and \(35n\) is \(1\), so it is prime.

Step6: Factor \(15x + 60y\)

Find the GCF of \(15x\) and \(60y\). The GCF of \(15\) and \(60\) is \(15\).
\(15x + 60y = 15(x + 4y)\)

Step7: Simplify \(5 + 2(-6y - 2) - 5y\)

First, distribute the \(2\): \(5 + 2(-6y) + 2(-2) - 5y = 5 - 12y - 4 - 5y\)
Then, combine like terms: \((5 - 4) + (-12y - 5y) = 1 - 17y\)

Step8: Simplify \(-1[x - (x - 3)] - 12 + 4x\)

First, simplify the inner parentheses: \(x - (x - 3) = x - x + 3 = 3\)
Then, distribute the \(-1\): \(-1(3) - 12 + 4x = -3 - 12 + 4x\)
Combine like terms: \((-3 - 12) + 4x = -15 + 4x = 4x - 15\)

Step9: Simplify \(9 - 4(2p + 3) - 8p + 7\)

First, distribute the \(-4\): \(9 - 8p - 12 - 8p + 7\)
Then, combine like terms: \((9 - 12 + 7) + (-8p - 8p) = 4 - 16p = -16p + 4 = -4(4p - 1)\)

Step10: Simplify \(-x + z(2(3x + y)) - 5y + 10x\)

First, distribute the \(z\) and \(2\): \(-x + 6xz + 2yz - 5y + 10x\)
Then, combine like terms: \((-x + 10x) + 6xz + 2yz - 5y = 9x + 6xz + 2yz - 5y\) (or factor further if possible, but this is simplified)

Answer:

1. \(2(7x - 10)\)
2. \(8(4y + 7)\)
3. prime
4. \(3(x + 3y)\)
5. prime
6. \(15(x + 4y)\)
7. \(1 - 17y\)
8. \(4x - 15\)
9. \(-4(4p - 1)\)
10. \(9x + 6xz + 2yz - 5y\) (or factored as \(x(9 + 6z) + y(2z - 5)\) but the simplified form is also acceptable)