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Difference of Squares practice questions

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Answer key with explanations

  1. Factor x2 − 81.

    Answer: (x + 9)(x − 9). 81 = 92, so a = x and b = 9, and x2 − 81 = (x + 9)(x − 9). The middle terms −9x and +9x cancel. The choice (x − 9)2 expands to x2 − 18x + 81, and (x + 9)2 expands to x2 + 18x + 81. Both have a middle term. A difference of squares does factor, so the last choice is wrong too.

  2. Which of these is a difference of squares?

    Answer: 36p2 − 25. A difference of squares has two perfect-square terms with a minus sign between them. 36p2 = (6p)2 and 25 = 52, so 36p2 − 25 = (6p + 5)(6p − 5). The second choice has a plus sign, so it is a sum of squares. In the third choice, 25p is not a perfect square because the exponent of p is 1. In the fourth, 36p is not a perfect square for the same reason.

  3. Factor 9x2 − 100.

    Answer: (3x + 10)(3x − 10). 9x2 = (3x)2 and 100 = 102, so a = 3x and b = 10, and the answer is (3x + 10)(3x − 10). Check: 9x2 − 30x + 30x − 100 = 9x2 − 100. The choice (9x + 10)(9x − 10) forgets to take the square root of 9. It expands to 81x2 − 100. The two squared choices have middle terms, −60x and +60x.

  4. Factor x8 − 25.

    Answer: (x4 + 5)(x4 − 5). Halve the exponent: x8 = (x4)2, because x4 · x4 = x8. Also 25 = 52. So a = x4 and b = 5, and the answer is (x4 + 5)(x4 − 5). The choice (x2 + 5)(x2 − 5) expands to x4 − 25, and (x8 + 5)(x8 − 5) expands to x16 − 25 (it forgot to take the square root of x8). Check at x = 2: 28 − 25 = 231 and (16 + 5)(16 − 5) = 21 × 11 = 231.

  5. Factor 12x2 − 27 completely, so that no factor can be factored further using integers.

    Answer: 3(2x + 3)(2x − 3). Neither 12 nor 27 is a perfect square, so take out the GCF 3 first: 12x2 − 27 = 3(4x2 − 9). Then 4x2 = (2x)2 and 9 = 32, so 4x2 − 9 = (2x + 3)(2x − 3). The full answer is 3(2x + 3)(2x − 3). The choice (2x + 3)(2x − 3) drops the 3 and equals only 4x2 − 9. The choice 3(2x − 3)2 has a middle term. The choice (6x + 9)(2x − 3) does multiply out to 12x2 − 27, but it is not completely factored, because 6x + 9 = 3(2x + 3) still has a common factor.

  6. Factor x4 − 256 completely, so that no factor can be factored further using integers.

    Answer: (x2 + 16)(x + 4)(x − 4). x4 = (x2)2 and 256 = 162, so x4 − 256 = (x2 + 16)(x2 − 16). The factor x2 − 16 = x2 − 42 factors again as (x + 4)(x − 4). The factor x2 + 16 is a sum of squares, so it stays. The complete answer is (x2 + 16)(x + 4)(x − 4). The second choice stops one step early. The third equals (x2 − 16)2 = x4 − 32x2 + 256. The fourth has a squared binomial, which makes a middle term. Check at x = 2: 24 − 256 = −240 and (4 + 16)(2 + 4)(2 − 4) = 20 × 6 × (−2) = −240.

  7. Only one of these binomials can be factored into two binomials with integer coefficients. Which one?

    Answer: 25y2 − 4. 25y2 − 4 = (5y)2 − 22 = (5y + 2)(5y − 2). The second and third choices are sums of squares, which are prime. In 25y2 − 21 the number 21 is not a perfect square, so the pattern does not apply, and there is no factoring with integers either.

  8. Use the difference-of-squares pattern to find 82 × 78.

    Answer: 6396. 82 = 80 + 2 and 78 = 80 − 2, so 82 × 78 = 802 − 22 = 6400 − 4 = 6396. The choice 6404 adds 22 instead of subtracting it, the choice 6400 forgets to subtract 22, and the choice 6392 subtracts 8 instead of 4.

  9. A square tile is 13 cm on each side. A square piece 7 cm on each side is cut off one corner, leaving an L-shaped tile. Which calculation gives the area of the L-shaped tile in square centimeters?

    Answer: (13 + 7)(13 − 7) = 20 × 6 = 120. The L-shape is the big square minus the small one: 132 − 72 = 169 − 49 = 120 square centimeters. By the pattern, 132 − 72 = (13 + 7)(13 − 7) = 20 × 6 = 120, which is the first choice. (The L can be cut and rearranged into a 20 cm by 6 cm rectangle.) The choices (13 − 7)2 and (13 + 7)2 are not 132 − 72. Adding the two squares counts the cut-off piece instead of removing it.

  10. Factor 98x2 − 8y2 completely, so that no factor can be factored further using integers.

    Answer: 2(7x + 2y)(7x − 2y). Take out the GCF 2 first: 98x2 − 8y2 = 2(49x2 − 4y2). Then 49x2 = (7x)2 and 4y2 = (2y)2, so 49x2 − 4y2 = (7x + 2y)(7x − 2y). The full answer is 2(7x + 2y)(7x − 2y). The second choice drops the 2, so it equals only 49x2 − 4y2. The third uses 4y instead of 2y: it forgot to take the square root of the 4 in 4y2, so it equals 2(49x2 − 16y2). The fourth has a squared binomial, which makes a middle term. Check at x = 2 and y = 1: 98(4) − 8 = 384 and 2(14 + 2)(14 − 2) = 2 × 16 × 12 = 384.

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