Factoring Trinomials (a & gt; 1) - Kuta Software - Free Printable

To factor each trinomial of the form ax² + bx + c (where a > 1), we use the "grouping method" or "splitting the middle term." Here’s how it works step by step for each problem:

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Problem 1: 3p² – 2p – 5

Step 1: Multiply a × c = 3 × (-5) = -15
Step 2: Find two numbers that multiply to -15 and add to b = -2 → those are -5 and +3
Step 3: Split the middle term:
3p² – 5p + 3p – 5
Step 4: Group: (3p² – 5p) + (3p – 5)
Step 5: Factor each group: p(3p – 5) + 1(3p – 5)
Step 6: Factor out common binomial: (p + 1)(3p – 5)

✔ Check: (p+1)(3p–5) = 3p² –5p +3p –5 = 3p² –2p –5 ✔️

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Problem 2: 2n² + 3n – 9

Step 1: a×c = 2×(-9) = -18
Step 2: Two numbers multiplying to -18, adding to 3 → +6 and -3
Step 3: Split: 2n² + 6n – 3n – 9
Step 4: Group: (2n² + 6n) + (-3n – 9)
Step 5: Factor: 2n(n + 3) –3(n + 3)
Step 6: Common factor: (2n – 3)(n + 3)

✔ Check: (2n–3)(n+3) = 2n² +6n –3n –9 = 2n² +3n –9 ✔️

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Problem 3: 3n² – 8n + 4

Step 1: a×c = 3×4 = 12
Step 2: Two numbers multiplying to 12, adding to -8 → -6 and -2
Step 3: Split: 3n² –6n –2n +4
Step 4: Group: (3n² –6n) + (-2n +4)
Step 5: Factor: 3n(n – 2) –2(n – 2)
Step 6: Common factor: (3n – 2)(n – 2)

✔ Check: (3n–2)(n–2) = 3n² –6n –2n +4 = 3n² –8n +4 ✔️

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Problem 4: 5n² + 19n + 12

Step 1: a×c = 5×12 = 60
Step 2: Two numbers multiplying to 60, adding to 19 → 15 and 4
Step 3: Split: 5n² +15n +4n +12
Step 4: Group: (5n² +15n) + (4n +12)
Step 5: Factor: 5n(n + 3) +4(n + 3)
Step 6: Common factor: (5n + 4)(n + 3)

✔ Check: (5n+4)(n+3) = 5n² +15n +4n +12 = 5n² +19n +12 ✔️

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Problem 5: 2v² + 11v + 5

Step 1: a×c = 2×5 = 10
Step 2: Two numbers multiplying to 10, adding to 11 → 10 and 1
Step 3: Split: 2v² +10v +v +5
Step 4: Group: (2v² +10v) + (v +5)
Step 5: Factor: 2v(v + 5) +1(v + 5)
Step 6: Common factor: (2v + 1)(v + 5)

✔ Check: (2v+1)(v+5) = 2v² +10v +v +5 = 2v² +11v +5 ✔️

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Problem 6: 2n² + 5n + 2

Step 1: a×c = 2×2 = 4
Step 2: Two numbers multiplying to 4, adding to 5 → 4 and 1
Step 3: Split: 2n² +4n +n +2
Step 4: Group: (2n² +4n) + (n +2)
Step 5: Factor: 2n(n + 2) +1(n + 2)
Step 6: Common factor: (2n + 1)(n + 2)

✔ Check: (2n+1)(n+2) = 2n² +4n +n +2 = 2n² +5n +2 ✔️

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Problem 7: 7a² + 53a + 28

Step 1: a×c = 7×28 = 196
Step 2: Two numbers multiplying to 196, adding to 53 → 49 and 4
Step 3: Split: 7a² +49a +4a +28
Step 4: Group: (7a² +49a) + (4a +28)
Step 5: Factor: 7a(a + 7) +4(a + 7)
Step 6: Common factor: (7a + 4)(a + 7)

✔ Check: (7a+4)(a+7) = 7a² +49a +4a +28 = 7a² +53a +28 ✔️

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Problem 8: 9k² + 66k + 21

Step 1: First, check if there’s a GCF. All terms divisible by 3?
9k² ÷3 = 3k², 66k÷3=22k, 21÷3=7 → so factor out 3 first:
3(3k² + 22k + 7)

Now factor inside: 3k² +22k +7
Step 2: a×c = 3×7 = 21
Step 3: Two numbers multiplying to 21, adding to 22 → 21 and 1
Step 4: Split: 3k² +21k +k +7
Step 5: Group: (3k² +21k) + (k +7)
Step 6: Factor: 3k(k + 7) +1(k + 7) → (3k + 1)(k + 7)

So full factorization: 3(3k + 1)(k + 7)

✔ Check: 3(3k+1)(k+7) = 3[3k² +21k +k +7] = 3[3k² +22k +7] = 9k² +66k +21 ✔️

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Final Answer:
1) (p + 1)(3p – 5)
2) (2n – 3)(n + 3)
3) (3n – 2)(n – 2)
4) (5n + 4)(n + 3)
5) (2v + 1)(v + 5)
6) (2n + 1)(n + 2)
7) (7a + 4)(a + 7)
8) 3(3k + 1)(k + 7)