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Factoring Out the Greatest Common Factor practice questions

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  1. Find the greatest common factor of 56 and 84.

    Answer: 28. Write the prime factors: 56 = 2 · 2 · 2 · 7 and 84 = 2 · 2 · 3 · 7. The shared part is 2 · 2 · 7 = 28. Check: 56 ÷ 28 = 2 and 84 ÷ 28 = 3, which share no factor larger than 1. The numbers 14 and 7 are common factors but not the greatest, and 168 is a multiple of both numbers, not a factor.

  2. Factor 32y + 56 completely.

    Answer: 8(4y + 7). The GCF of 32 and 56 is 8. Divide each term by 8: 32y ÷ 8 = 4y and 56 ÷ 8 = 7, so the answer is 8(4y + 7). Check: 8 · 4y + 8 · 7 = 32y + 56. The choice 4(8y + 14) is correct but not complete, because 8 and 14 still share the factor 2. The choice 8(4y + 56) forgets to divide the 56 by 8, and 8(y + 7) forgets to divide the 32 by 8.

  3. Which of these expressions has no common factor greater than 1 that you can factor out?

    Answer: 8x + 9. Check each expression. 10x + 15 has the common factor 5, 9x2 + 6x has the common factor 3x, and 4x2 − 12 has the common factor 4. In 8x + 9, the numbers 8 and 9 share only the factor 1, and 9 has no x, so there is no common factor to pull out.

  4. Find the greatest common factor of 20c5d3 and 30c2d4.

    Answer: 10c2d3. The GCF of the coefficients 20 and 30 is 10. The letter c has exponents 5 and 2, so use the lowest, c2. The letter d has exponents 3 and 4, so use the lowest, d3. The GCF is 10c2d3. Check: 20c5d3 ÷ (10c2d3) = 2c3 and 30c2d4 ÷ (10c2d3) = 3d, which share no common factor. The choice 10c5d4 uses the highest exponents, which do not divide both terms. The choice 5c2d3 is a common factor but not the greatest, and 10cd stops at exponent 1 instead of using the lowest exponents.

  5. Factor 20y4 − 35y2 completely.

    Answer: 5y2(4y2 − 7). The GCF of 20 and 35 is 5, and the lowest exponent of y is 2, so the GCF is 5y2. Divide: 20y4 ÷ (5y2) = 4y2 and −35y2 ÷ (5y2) = −7. The answer is 5y2(4y2 − 7). Check: 5y2 · 4y2 = 20y4 and 5y2 · (−7) = −35y2. The choices 5y(4y3 − 7y) and 5(4y4 − 7y2) multiply out correctly but stop early, because a common factor is still inside. The choice 5y2(4y2 + 7) has the wrong sign on the second term.

  6. Factor −9n2 − 12n by taking out the negative of the greatest common factor.

    Answer: −3n(3n + 4). The GCF of 9 and 12 is 3, and the lowest exponent of n is 1, so the GCF is 3n. Factor out its negative, −3n, and divide each term by it: −9n2 ÷ (−3n) = 3n and −12n ÷ (−3n) = 4. The answer is −3n(3n + 4). Check: −3n · 3n = −9n2 and −3n · 4 = −12n. The choice 3n(−3n − 4) equals the original but takes out the positive GCF. The choice −3n(3n − 4) has a wrong sign (it gives +12n). The choice −3(3n2 + 4n) takes out only the number, not the full GCF.

  7. Factor 14w3 + 7w2 completely.

    Answer: 7w2(2w + 1). The GCF of 14 and 7 is 7, and the lowest exponent of w is 2, so the GCF is 7w2. Divide: 14w3 ÷ (7w2) = 2w and 7w2 ÷ (7w2) = 1. A term that equals the GCF leaves a 1, so the answer is 7w2(2w + 1). Check: 7w2 · 2w = 14w3 and 7w2 · 1 = 7w2. The choice 7w2(2w) forgets the 1 and multiplies out to only 14w3. The choices 7w(2w2 + w) and 7(2w3 + w2) are correct but stop early, because a common factor is still inside.

  8. Factor 16m4 − 24m3 + 40m2 completely.

    Answer: 8m2(2m2 − 3m + 5). The GCF of 16, 24 and 40 is 8, and the lowest exponent of m is 2, so the GCF is 8m2. Divide each term: 16m4 ÷ (8m2) = 2m2, −24m3 ÷ (8m2) = −3m and 40m2 ÷ (8m2) = 5. The answer is 8m2(2m2 − 3m + 5). Check: 8m2 · 2m2 = 16m4, 8m2 · (−3m) = −24m3 and 8m2 · 5 = 40m2. The choices 8m(2m3 − 3m2 + 5m) and 4m2(4m2 − 6m + 10) multiply out correctly but are not complete (an m is still common inside the first, and a 2 inside the second). The choice 8m2(2m2 + 3m + 5) has the wrong sign on the middle term.

  9. Factor 4y(y − 3) + 7(y − 3).

    Answer: (y − 3)(4y + 7). Both terms contain the binomial (y − 3), so pull it out. What is left from the first term is 4y and from the second is 7, so the answer is (y − 3)(4y + 7). Check by expanding: (y − 3)(4y + 7) = 4y2 + 7y − 12y − 21 = 4y2 − 5y − 21, and 4y(y − 3) + 7(y − 3) = 4y2 − 12y + 7y − 21 = 4y2 − 5y − 21. The choice (y − 3)(4y − 7) changes the sign of the 7. The choice 11y(y − 3) adds the leftovers 4y and 7 as if they were like terms. The choice (y + 3)(4y + 7) changes the sign inside the common binomial.

  10. A rectangular patio has an area of 20x2 + 15x square feet, and one side is 5x feet long. Which expression gives the length of the other side, in feet? (Assume x > 0.)

    Answer: 4x + 3. Area equals length times width, so the other side is the area divided by 5x. Factor out 5x: 20x2 + 15x = 5x(4x + 3), so the other side is 4x + 3 feet. Check: 5x · (4x + 3) = 20x2 + 15x. The choice 4x + 15 divides only the first term by 5x. The choice 4x2 + 3x divides the numbers by 5 but leaves the letters alone. The choice 20x2 + 10x subtracts 5x from the area, but a side length comes from dividing, not subtracting.

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