➕ Algebra 1 · Factoring

Factoring Out the Greatest Common Factor: true or false

Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.

All statements and answers

• The GCF of 54 and 72 is 18.

  ✓ True. 54 = 2 · 3 · 3 · 3 and 72 = 2 · 2 · 2 · 3 · 3. The shared part is 2 · 3 · 3 = 18. Then 54 ÷ 18 = 3 and 72 ÷ 18 = 4, which share no factor larger than 1.

• The greatest common factor of 8 and 12 is 24.

  ✗ False. 24 is a multiple of both numbers, not a factor of them. A factor must divide both numbers, so the GCF is 4.

• The GCF of 5x³ and 10x is 5x³.

  ✗ False. 5x³ does not divide 10x evenly, because the exponent 3 is bigger than 1. Use the lowest exponent: the GCF is 5x, and 5x³ + 10x = 5x(x² + 2).

• The GCF of 6a² and 9b is 3ab.

  ✗ False. Neither letter is in both terms: a is missing from 9b and b is missing from 6a². The GCF is just 3.

• 10p² + 2p = 2p(5p).

  ✗ False. The term 2p equals the GCF, so it leaves a 1: 2p ÷ (2p) = 1. The correct factoring is 2p(5p + 1). Distributing 2p(5p) gives only 10p², so the second term is lost.

• −6x² + 15x = −3x(2x + 5).

  ✗ False. Distributing gives −3x · 5 = −15x, not +15x. Dividing 15x by −3x gives −5, so the correct factoring is −3x(2x − 5).

• Both 2y(−5y + 2) and −2y(5y − 2) equal −10y² + 4y.

  ✓ True. 2y · (−5y) = −10y² and 2y · 2 = 4y. Also −2y · 5y = −10y² and −2y · (−2) = 4y. Both are correct. The second, with the negative GCF, is the usual convention.

• If the GCF of the terms is 1, the expression cannot be factored in any way.

  ✗ False. A GCF of 1 only means there is no common factor to pull out. For example, x² + 8x + 12 has GCF 1, but it equals (x + 2)(x + 6).

• In 5x(x + 3) + 2(x + 3), the binomial (x + 3) is a common factor, so the expression equals (x + 3)(5x + 2).

  ✓ True. Pulling out (x + 3) leaves 5x and 2. Check by expanding both: (x + 3)(5x + 2) = 5x² + 17x + 6 and 5x(x + 3) + 2(x + 3) = 5x² + 15x + 2x + 6 = 5x² + 17x + 6.

• 3(x² + 2x) is a completely factored form of 3x² + 6x.

  ✗ False. It multiplies out correctly, but x² + 2x still has the common factor x. Taking out the full GCF, 3x, gives 3x(x + 2).

• Distributing your answer and getting the original expression shows the factoring is correct, but it does not show the factoring is complete.

  ✓ True. For example, 4(2x + 6) distributes to 8x + 24, so it is correct, but it is not complete because 2x + 6 still has a common factor of 2. The complete form is 8(x + 3).

• The terms 12x⁴ and 18x² have GCF 6x².

  ✓ True. The GCF of 12 and 18 is 6, and the lowest exponent of x is 2. Check: 12x⁴ ÷ (6x²) = 2x² and 18x² ÷ (6x²) = 3, which share no common factor greater than 1.

• 4a³ + 8a² + 6a = 2a(2a² + 4a + 3).

  ✓ True. The GCF of 4, 8 and 6 is 2, and the lowest exponent of a is 1. Distributing gives 2a · 2a² = 4a³, 2a · 4a = 8a² and 2a · 3 = 6a.