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Factoring by Grouping practice questions

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Answer key with explanations

  1. Which of these is the factored form of x3 + 4x2 + 3x + 12, written as a product?

    Answer: (x + 4)(x2 + 3). Pair the terms: (x3 + 4x2) + (3x + 12) = x2(x + 4) + 3(x + 4). Both pairs leave (x + 4), so pull it out: (x + 4)(x2 + 3). The second choice has the right pieces but is still a sum of two terms, not a product. The third has the wrong sign: it expands to x3 + 4x2 − 3x − 12. The fourth swaps the pieces: it expands to x3 + 3x2 + 4x + 12.

  2. In x3 − 5x2 − 2x + 10, grouping the first two terms gives x2(x − 5). What should you take out of the second pair, −2x + 10, so that it leaves the same binomial, (x − 5)?

    Answer: −2. Take out −2: (−2)(x) = −2x and (−2)(−5) = +10, so −2x + 10 = −2(x − 5). Both pairs now leave (x − 5), and x3 − 5x2 − 2x + 10 = x2(x − 5) − 2(x − 5) = (x − 5)(x2 − 2). Taking out +2 gives 2(−x + 5), whose binomial is the opposite of (x − 5). The choices −10 and 10 do not divide −2x evenly with integer coefficients.

  3. Factor x3 − 6x2 − 5x + 30.

    Answer: (x − 6)(x2 − 5). Pair the terms: (x3 − 6x2) + (−5x + 30) = x2(x − 6) − 5(x − 6) = (x − 6)(x2 − 5). The second pair needs −5, because −5x + 30 = −5(x − 6). The choice (x − 6)(x2 + 5) comes from taking out +5 by mistake, and it expands to x3 − 6x2 + 5x − 30. The choice (x + 6)(x2 − 5) expands to x3 + 6x2 − 5x − 30, and (x − 5)(x2 − 6) expands to x3 − 5x2 − 6x + 30.

  4. Factor 2x3 + 8x2 − 6x − 24 completely, so that no factor can be factored further using integers.

    Answer: 2(x + 4)(x2 − 3). Take out the GCF 2 first: 2(x3 + 4x2 − 3x − 12). Group inside: x2(x + 4) − 3(x + 4) = (x + 4)(x2 − 3). The complete answer is 2(x + 4)(x2 − 3). The choice without the 2 equals only x3 + 4x2 − 3x − 12, which is half of the original. The choice with x2 + 3 has the wrong sign. The choice (2x + 8)(x2 − 3) does multiply out to the original, but 2x + 8 = 2(x + 4) still has a common factor, so it is not completely factored.

  5. Factor 3xy + 2 + 6x + y. (The terms are not in a helpful order, so rearrange them first.)

    Answer: (y + 2)(3x + 1). Rearrange so that terms with a common factor are next to each other: 3xy + 6x + y + 2. Pair: 3x(y + 2) + 1(y + 2) = (y + 2)(3x + 1). The second pair y + 2 has GCF 1, so it leaves 1(y + 2), which is easy to drop by mistake. Dropping it gives 3x(y + 2), which equals only 3xy + 6x. Check the correct answer: (y + 2)(3x + 1) = 3xy + y + 6x + 2. The choice (y + 2)(3x + 2) expands to 3xy + 2y + 6x + 4, and (y + 1)(3x + 2) expands to 3xy + 2y + 3x + 2.

  6. Only one of these polynomials can be factored by pairing the first two terms and the last two terms and pulling out a common binomial. Which one?

    Answer: x3 + 5x2 + 2x + 10. In the first, (x3 + 5x2) + (2x + 10) = x2(x + 5) + 2(x + 5) = (x + 5)(x2 + 2). In the others, the first pair always leaves (x + 5), but the second pair leaves (x + 6), (x + 4) or (x − 5) after taking out 2 (from 2x + 12, 2x + 8 and 2x − 10). None of these matches (x + 5), and taking out −2 from 2x − 10 gives −2(−x + 5), which still does not match.

  7. To factor 3x2 + 11x + 6 by grouping, which way of splitting 11x works?

    Answer: 9x + 2x. Look for two numbers with product a × c = 3 × 6 = 18 and sum 11. Since 9 × 2 = 18 and 9 + 2 = 11, split 11x = 9x + 2x. Then 3x2 + 9x + 2x + 6 = 3x(x + 3) + 2(x + 3) = (x + 3)(3x + 2). The other choices all add to 11, but their products are 30, 24 and 10, not 18, so their pairs do not leave matching binomials.

  8. Factor 4x2 − 9x − 9.

    Answer: (x − 3)(4x + 3). Here a × c = 4 × (−9) = −36 and b = −9. The numbers −12 and 3 have product −36 and sum −9. Split: 4x2 − 12x + 3x − 9 = 4x(x − 3) + 3(x − 3) = (x − 3)(4x + 3). Check: (x − 3)(4x + 3) = 4x2 + 3x − 12x − 9. The other choices expand to 4x2 + 9x − 9, 4x2 − 15x + 9 and 4x2 − 9.

  9. Factor 2x3 − 2x2 − 18x + 18 completely, so that no factor can be factored further using integers.

    Answer: 2(x − 1)(x + 3)(x − 3). Take out the GCF 2: 2(x3 − x2 − 9x + 9). Group inside: x2(x − 1) − 9(x − 1) = (x − 1)(x2 − 9). Then x2 − 9 is a difference of squares, (x + 3)(x − 3). The complete answer is 2(x − 1)(x + 3)(x − 3). The choice 2(x − 1)(x2 − 9) stops one step early. The choice without the 2 equals half of the original. The choice with (x + 1) has the wrong sign. Check at x = 2: 2(8) − 2(4) − 36 + 18 = −10 and 2(1)(5)(−1) = −10.

  10. A theater section has x3 + 6x2 + 2x + 12 seats arranged in equal rows. There are x + 6 rows. How many seats are in each row?

    Answer: x2 + 2. The total number of seats is the number of rows times the seats in each row, so the seats per row is the other factor. Group: (x3 + 6x2) + (2x + 12) = x2(x + 6) + 2(x + 6) = (x + 6)(x2 + 2). One factor is the number of rows, x + 6, so the other factor, x2 + 2, is the number of seats in each row. Check: (x + 6)(x2 + 2) = x3 + 2x + 6x2 + 12. For example, with x = 4 there are 64 + 96 + 8 + 12 = 180 seats and 10 rows, which is 18 = 42 + 2 seats per row.

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