Free Printable Factoring Trinomials Worksheets [PDFs] - Free Printable

Let's solve each trinomial factoring problem step by step. The goal is to factor each quadratic trinomial completely.

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1) \( 7m^2 + 6m - 1 \)

We look for two numbers that multiply to \( 7 \times (-1) = -7 \) and add to \( 6 \).
Try: \( 7 \) and \( -1 \): \( 7 \times (-1) = -7 \), \( 7 + (-1) = 6 \) → Yes!

Split the middle term:
\[
7m^2 + 7m - m - 1
\]
Group:
\[
(7m^2 + 7m) - (m + 1) = 7m(m + 1) -1(m + 1)
\]
Factor:
\[
(7m - 1)(m + 1)
\]

✔ Answer: \( (7m - 1)(m + 1) \)

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2) \( 3k^2 - 10k + 7 \)

Find two numbers that multiply to \( 3 \times 7 = 21 \), add to \( -10 \).
Try: \( -3 \) and \( -7 \): \( -3 \times -7 = 21 \), \( -3 + (-7) = -10 \) → Yes!

Split:
\[
3k^2 - 3k - 7k + 7
\]
Group:
\[
3k(k - 1) -7(k - 1) = (3k - 7)(k - 1)
\]

✔ Answer: \( (3k - 7)(k - 1) \)

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3) \( 5j^2 - 36j - 81 \)

Multiply: \( 5 \times (-81) = -405 \), need two numbers adding to \( -36 \).
Try: \( -45 \) and \( 9 \): \( -45 \times 9 = -405 \), \( -45 + 9 = -36 \)

Split:
\[
5j^2 - 45j + 9j - 81
\]
Group:
\[
5j(j - 9) + 9(j - 9) = (5j + 9)(j - 9)
\]

✔ Answer: \( (5j + 9)(j - 9) \)

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4) \( 2x^2 - 9x - 81 \)

\( 2 \times (-81) = -162 \), need sum \( -9 \).
Try: \( -18 \) and \( 9 \): \( -18 \times 9 = -162 \), \( -18 + 9 = -9 \)

Split:
\[
2x^2 - 18x + 9x - 81
\]
Group:
\[
2x(x - 9) + 9(x - 9) = (2x + 9)(x - 9)
\]

✔ Answer: \( (2x + 9)(x - 9) \)

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5) \( 3n^2 - 16n + 20 \)

\( 3 \times 20 = 60 \), need sum \( -16 \).
Try: \( -10 \) and \( -6 \): \( -10 \times -6 = 60 \), \( -10 + (-6) = -16 \)

Split:
\[
3n^2 - 10n - 6n + 20
\]
Group:
\[
n(3n - 10) -2(3n - 10) = (n - 2)(3n - 10)
\]

✔ Answer: \( (n - 2)(3n - 10) \)

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6) \( 2r^2 + 7r - 30 \)

\( 2 \times (-30) = -60 \), sum \( 7 \).
Try: \( 12 \) and \( -5 \): \( 12 \times -5 = -60 \), \( 12 + (-5) = 7 \)

Split:
\[
2r^2 + 12r - 5r - 30
\]
Group:
\[
2r(r + 6) -5(r + 6) = (2r - 5)(r + 6)
\]

✔ Answer: \( (2r - 5)(r + 6) \)

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7) \( 5k^2 + 8k + 80 \)

Wait — check discriminant: \( b^2 - 4ac = 64 - 4(5)(80) = 64 - 1600 = -1536 \) → negative, so no real factors.

But let’s double-check if it can be factored.

Wait — perhaps typo? Let's see: \( 5k^2 + 8k + 80 \)

No integers work: \( 5 \times 80 = 400 \), need two numbers with product 400 and sum 8 → not possible.

Try: \( 5k^2 + 8k + 80 \) → likely prime (not factorable over integers).

✔ Answer: Prime (cannot be factored over integers)

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8) \( 5x^2 - 14x + 8 \)

\( 5 \times 8 = 40 \), need sum \( -14 \). Try: \( -10 \) and \( -4 \): \( -10 \times -4 = 40 \), \( -10 + (-4) = -14 \)

Split:
\[
5x^2 - 10x - 4x + 8
\]
Group:
\[
5x(x - 2) -4(x - 2) = (5x - 4)(x - 2)
\]

✔ Answer: \( (5x - 4)(x - 2) \)

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9) \( 7p^2 - 20p + 12 \)

\( 7 \times 12 = 84 \), sum \( -20 \). Try: \( -14 \) and \( -6 \): \( -14 \times -6 = 84 \), sum \( -20 \)

Split:
\[
7p^2 - 14p - 6p + 12
\]
Group:
\[
7p(p - 2) -6(p - 2) = (7p - 6)(p - 2)
\]

✔ Answer: \( (7p - 6)(p - 2) \)

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10) \( 3v^2 + 14v - 49 \)

\( 3 \times (-49) = -147 \), sum \( 14 \). Try: \( 21 \) and \( -7 \): \( 21 \times -7 = -147 \), sum \( 14 \)

Split:
\[
3v^2 + 21v - 7v - 49
\]
Group:
\[
3v(v + 7) -7(v + 7) = (3v - 7)(v + 7)
\]

✔ Answer: \( (3v - 7)(v + 7) \)

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11) \( 7t^2 - 26t - 45 \)

\( 7 \times (-45) = -315 \), sum \( -26 \). Try: \( -35 \) and \( 9 \): \( -35 \times 9 = -315 \), sum \( -26 \)

Split:
\[
7t^2 - 35t + 9t - 45
\]
Group:
\[
7t(t - 5) + 9(t - 5) = (7t + 9)(t - 5)
\]

✔ Answer: \( (7t + 9)(t - 5) \)

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12) \( 5p^2 - 52p + 20 \)

\( 5 \times 20 = 100 \), sum \( -52 \). Try: \( -50 \) and \( -2 \): \( -50 \times -2 = 100 \), sum \( -52 \)

Split:
\[
5p^2 - 50p - 2p + 20
\]
Group:
\[
5p(p - 10) -2(p - 10) = (5p - 2)(p - 10)
\]

✔ Answer: \( (5p - 2)(p - 10) \)

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13) \( 5s^2 - 43s + 24 \)

\( 5 \times 24 = 120 \), sum \( -43 \). Try: \( -40 \) and \( -3 \): \( -40 \times -3 = 120 \), sum \( -43 \)

Split:
\[
5s^2 - 40s - 3s + 24
\]
Group:
\[
5s(s - 8) -3(s - 8) = (5s - 3)(s - 8)
\]

✔ Answer: \( (5s - 3)(s - 8) \)

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14) \( 5f^2 + 26f + 24 \)

\( 5 \times 24 = 120 \), sum \( 26 \). Try: \( 20 \) and \( 6 \): \( 20 \times 6 = 120 \), sum \( 26 \)

Split:
\[
5f^2 + 20f + 6f + 24
\]
Group:
\[
5f(f + 4) + 6(f + 4) = (5f + 6)(f + 4)
\]

✔ Answer: \( (5f + 6)(f + 4) \)

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15) \( 3r^2 + 40r + 100 \)

\( 3 \times 100 = 300 \), sum \( 40 \). Try: \( 30 \) and \( 10 \): \( 30 \times 10 = 300 \), sum \( 40 \)

Split:
\[
3r^2 + 30r + 10r + 100
\]
Group:
\[
3r(r + 10) + 10(r + 10) = (3r + 10)(r + 10)
\]

✔ Answer: \( (3r + 10)(r + 10) \)

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16) \( 2x^2 - 3x - 5 \)

\( 2 \times (-5) = -10 \), sum \( -3 \). Try: \( -5 \) and \( 2 \): \( -5 \times 2 = -10 \), sum \( -3 \)

Split:
\[
2x^2 - 5x + 2x - 5
\]
Group:
\[
x(2x - 5) + 1(2x - 5) = (x + 1)(2x - 5)
\]

✔ Answer: \( (x + 1)(2x - 5) \)

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17) \( 5p^2 + 19p + 12 \)

\( 5 \times 12 = 60 \), sum \( 19 \). Try: \( 15 \) and \( 4 \): \( 15 \times 4 = 60 \), sum \( 19 \)

Split:
\[
5p^2 + 15p + 4p + 12
\]
Group:
\[
5p(p + 3) + 4(p + 3) = (5p + 4)(p + 3)
\]

✔ Answer: \( (5p + 4)(p + 3) \)

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18) \( 2m^2 + 3m - 27 \)

\( 2 \times (-27) = -54 \), sum \( 3 \). Try: \( 9 \) and \( -6 \): \( 9 \times -6 = -54 \), sum \( 3 \)

Split:
\[
2m^2 + 9m - 6m - 27
\]
Group:
\[
m(2m + 9) -3(2m + 9) = (m - 3)(2m + 9)
\]

✔ Answer: \( (m - 3)(2m + 9) \)

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19) \( 3n^2 + 10n - 8 \)

\( 3 \times (-8) = -24 \), sum \( 10 \). Try: \( 12 \) and \( -2 \): \( 12 \times -2 = -24 \), sum \( 10 \)

Split:
\[
3n^2 + 12n - 2n - 8
\]
Group:
\[
3n(n + 4) -2(n + 4) = (3n - 2)(n + 4)
\]

✔ Answer: \( (3n - 2)(n + 4) \)

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20) \( 2d^2 + 7d - 7 \)

\( 2 \times (-7) = -14 \), sum \( 7 \). Try: \( 14 \) and \( -1 \): \( 14 \times -1 = -14 \), sum \( 13 \) → no.

Try: \( 7 \) and \( -2 \): \( 7 \times -2 = -14 \), sum \( 5 \) → no.

Check discriminant: \( 49 - 4(2)(-7) = 49 + 56 = 105 \) → not a perfect square → not factorable over integers.

✔ Answer: Prime

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21) \( 10b^2 - 21b - 49 \)

\( 10 \times (-49) = -490 \), sum \( -21 \). Try: \( -35 \) and \( 14 \): \( -35 \times 14 = -490 \), sum \( -21 \)

Split:
\[
10b^2 - 35b + 14b - 49
\]
Group:
\[
5b(2b - 7) + 7(2b - 7) = (5b + 7)(2b - 7)
\]

✔ Answer: \( (5b + 7)(2b - 7) \)

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22) \( 6x^2 + 41x + 70 \)

\( 6 \times 70 = 420 \), sum \( 41 \). Try: \( 21 \) and \( 20 \): \( 21 \times 20 = 420 \), sum \( 41 \)

Split:
\[
6x^2 + 21x + 20x + 70
\]
Group:
\[
3x(2x + 7) + 10(2x + 7) = (3x + 10)(2x + 7)
\]

✔ Answer: \( (3x + 10)(2x + 7) \)

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23) \( 9z^2 + 9z - 40 \)

\( 9 \times (-40) = -360 \), sum \( 9 \). Try: \( 24 \) and \( -15 \): \( 24 \times -15 = -360 \), sum \( 9 \)

Split:
\[
9z^2 + 24z - 15z - 40
\]
Group:
\[
3z(3z + 8) -5(3z + 8) = (3z - 5)(3z + 8)
\]

✔ Answer: \( (3z - 5)(3z + 8) \)

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24) \( 8n^2 + 71n - 90 \)

\( 8 \times (-90) = -720 \), sum \( 71 \). Try: \( 80 \) and \( -9 \): \( 80 \times -9 = -720 \), sum \( 71 \)

Split:
\[
8n^2 + 80n - 9n - 90
\]
Group:
\[
8n(n + 10) -9(n + 10) = (8n - 9)(n + 10)
\]

✔ Answer: \( (8n - 9)(n + 10) \)

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25) \( 4m^2 - 4m - 63 \)

\( 4 \times (-63) = -252 \), sum \( -4 \). Try: \( -18 \) and \( 14 \): \( -18 \times 14 = -252 \), sum \( -4 \)

Split:
\[
4m^2 - 18m + 14m - 63
\]
Group:
\[
2m(2m - 9) + 7(2m - 9) = (2m + 7)(2m - 9)
\]

✔ Answer: \( (2m + 7)(2m - 9) \)

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26) \( 6t^2 + 37t + 45 \)

\( 6 \times 45 = 270 \), sum \( 37 \). Try: \( 30 \) and \( 7 \): \( 30 \times 7 = 210 \) → no.

Try: \( 27 \) and \( 10 \): \( 27 \times 10 = 270 \), sum \( 37 \) → yes!

Split:
\[
6t^2 + 27t + 10t + 45
\]
Group:
\[
3t(2t + 9) + 5(2t + 9) = (3t + 5)(2t + 9)
\]

✔ Answer: \( (3t + 5)(2t + 9) \)

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✔ Final Answers:

| Problem | Factored Form |
|--------|----------------|
| 1 | \( (7m - 1)(m + 1) \) |
| 2 | \( (3k - 7)(k - 1) \) |
| 3 | \( (5j + 9)(j - 9) \) |
| 4 | \( (2x + 9)(x - 9) \) |
| 5 | \( (n - 2)(3n - 10) \) |
| 6 | \( (2r - 5)(r + 6) \) |
| 7 | Prime |
| 8 | \( (5x - 4)(x - 2) \) |
| 9 | \( (7p - 6)(p - 2) \) |
| 10 | \( (3v - 7)(v + 7) \) |
| 11 | \( (7t + 9)(t - 5) \) |
| 12 | \( (5p - 2)(p - 10) \) |
| 13 | \( (5s - 3)(s - 8) \) |
| 14 | \( (5f + 6)(f + 4) \) |
| 15 | \( (3r + 10)(r + 10) \) |
| 16 | \( (x + 1)(2x - 5) \) |
| 17 | \( (5p + 4)(p + 3) \) |
| 18 | \( (m - 3)(2m + 9) \) |
| 19 | \( (3n - 2)(n + 4) \) |
| 20 | Prime |
| 21 | \( (5b + 7)(2b - 7) \) |
| 22 | \( (3x + 10)(2x + 7) \) |
| 23 | \( (3z - 5)(3z + 8) \) |
| 24 | \( (8n - 9)(n + 10) \) |
| 25 | \( (2m + 7)(2m - 9) \) |
| 26 | \( (3t + 5)(2t + 9) \) |

> 💡 Note: For problems where no integer pairs work (e.g., #7 and #20), we conclude they are prime (not factorable over integers).

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