Question

bell: ______ homework 8: factoring trinomials (ax² + bx directions: factor each polynomial. check your answer by distributing. 1. 2x² + 5x + 2 2. 3n² + 5n + 2 3. 2y² + 9y - 5 4. 3g² - 7g + 2 5. 2k² - 11k + 15 6. 9p² + 6p - 8 7. 6m² - 13m + 6 8. 3c² + 2c - 16 9. 4b² + 12b + 9 directions: factor each polynomial. look for a gcf first. 10. 8x² - 2x - 10 11. 6a² + 9a - 27 12. 60x² + 4x - 8 13. 16x² - 24x + 8 14. 20x² + 55x + 30 15. 10x² - 32x + 24 © gina wilson (all things algebra®, llc), 2012 - 201

Explanation:

Step1: Factor \(2x^2 + 5x + 2\)

We need two numbers that multiply to \(2\times2 = 4\) and add up to \(5\). The numbers are \(4\) and \(1\).
Rewrite the middle term: \(2x^2 + 4x + x + 2\)
Group the terms: \((2x^2 + 4x) + (x + 2)\)
Factor out the GCF from each group: \(2x(x + 2) + 1(x + 2)\)
Factor out \((x + 2)\): \((2x + 1)(x + 2)\)

Step2: Factor \(3n^2 + 5n + 2\)

Multiply \(3\times2 = 6\). Find two numbers that multiply to \(6\) and add to \(5\) (numbers \(3\) and \(2\)).
Rewrite: \(3n^2 + 3n + 2n + 2\)
Group: \((3n^2 + 3n) + (2n + 2)\)
Factor: \(3n(n + 1) + 2(n + 1)=(3n + 2)(n + 1)\)

Step3: Factor \(2y^2 + 9y - 5\)

Multiply \(2\times(-5)= -10\). Find two numbers that multiply to \(-10\) and add to \(9\) (numbers \(10\) and \(-1\)).
Rewrite: \(2y^2 + 10y - y - 5\)
Group: \((2y^2 + 10y) + (-y - 5)\)
Factor: \(2y(y + 5) - 1(y + 5)=(2y - 1)(y + 5)\)

Step4: Factor \(3g^2 - 7g + 2\)

Multiply \(3\times2 = 6\). Numbers that multiply to \(6\) and add to \(-7\) (wait, sum is \(-7\), product \(6\): \(-6\) and \(-1\)).
Rewrite: \(3g^2 - 6g - g + 2\)
Group: \((3g^2 - 6g) + (-g + 2)\)
Factor: \(3g(g - 2) - 1(g - 2)=(3g - 1)(g - 2)\)

Step5: Factor \(2k^2 - 11k + 15\)

Multiply \(2\times15 = 30\). Numbers that multiply to \(30\) and add to \(-11\) (\(-6\) and \(-5\)).
Rewrite: \(2k^2 - 6k - 5k + 15\)
Group: \((2k^2 - 6k) + (-5k + 15)\)
Factor: \(2k(k - 3) - 5(k - 3)=(2k - 5)(k - 3)\)

Step6: Factor \(9p^2 + 6p - 8\)

Multiply \(9\times(-8)= -72\). Numbers that multiply to \(-72\) and add to \(6\) ( \(12\) and \(-6\)).
Rewrite: \(9p^2 + 12p - 6p - 8\)
Group: \((9p^2 + 12p) + (-6p - 8)\)
Factor: \(3p(3p + 4) - 2(3p + 4)=(3p - 2)(3p + 4)\)

Step7: Factor \(6m^2 - 13m + 6\)

Multiply \(6\times6 = 36\). Numbers that multiply to \(36\) and add to \(-13\) (\(-9\) and \(-4\)).
Rewrite: \(6m^2 - 9m - 4m + 6\)
Group: \((6m^2 - 9m) + (-4m + 6)\)
Factor: \(3m(2m - 3) - 2(2m - 3)=(3m - 2)(2m - 3)\)

Step8: Factor \(3c^2 + 2c - 16\)

Multiply \(3\times(-16)= -48\). Numbers that multiply to \(-48\) and add to \(2\) ( \(8\) and \(-6\)).
Rewrite: \(3c^2 + 8c - 6c - 16\)
Group: \((3c^2 + 8c) + (-6c - 16)\)
Factor: \(c(3c + 8) - 2(3c + 8)=(c - 2)(3c + 8)\)

Step9: Factor \(4b^2 + 12b + 9\)

This is a perfect square trinomial: \((2b)^2 + 2\times2b\times3 + 3^2=(2b + 3)^2\)

Step10: Factor \(8x^2 - 2x - 10\) (GCF first: GCF of \(8, -2, -10\) is \(2\))

Factor out \(2\): \(2(4x^2 - x - 5)\)
Now factor \(4x^2 - x - 5\). Multiply \(4\times(-5)= -20\). Numbers \(4\) and \(-5\) (add to \(-1\)).
Rewrite: \(4x^2 + 4x - 5x - 5\)
Group: \((4x^2 + 4x) + (-5x - 5)\)
Factor: \(4x(x + 1) - 5(x + 1)=(4x - 5)(x + 1)\)
So overall: \(2(4x - 5)(x + 1)\)

Step11: Factor \(6a^2 + 9a - 27\) (GCF is \(3\))

Factor out \(3\): \(3(2a^2 + 3a - 9)\)
Factor \(2a^2 + 3a - 9\). Multiply \(2\times(-9)= -18\). Numbers \(6\) and \(-3\) (add to \(3\)).
Rewrite: \(2a^2 + 6a - 3a - 9\)
Group: \((2a^2 + 6a) + (-3a - 9)\)
Factor: \(2a(a + 3) - 3(a + 3)=(2a - 3)(a + 3)\)
Overall: \(3(2a - 3)(a + 3)\)

Step12: Factor \(60x^2 + 4x - 8\) (GCF is \(4\))

Factor out \(4\): \(4(15x^2 + x - 2)\)
Factor \(15x^2 + x - 2\). Multiply \(15\times(-2)= -30\). Numbers \(6\) and \(-5\) (add to \(1\)).
Rewrite: \(15x^2 + 6x - 5x - 2\)
Group: \((15x^2 + 6x) + (-5x - 2)\)
Factor: \(3x(5x + 2) - 1(5x + 2)=(3x - 1)(5x + 2)\)
Overall: \(4(3x - 1)(5x + 2)\)

Step13: Factor \(16x^2 - 24x + 8\) (GCF is \(8\))

Factor out \(8\): \(8(2x^2 - 3x + 1)\)
Factor \(2x^2 - 3x + 1\). Multiply \(2\times1 = 2\). Numbers \(-2\) and \(-1\) (add to \(-3\)).
Rewrite: \(2x^2 - 2x - x + 1\)
Group: \((2x^2 - 2x) + (-x + 1)\)
Factor: \(2x(x - 1) - 1(x - 1)=(2x - 1)(x - 1)\)
Overall: \(8(2x - 1)(x - 1)\)

Step14: Factor \(20x^2 + 55x + 30\) (GCF is \(5\))

Factor out \(5\): \(5(4x^2 + 11x + 6)\)
Factor \(4x^2 + 11x + 6\). Multiply \(4\times6 = 24\). Numbers \(8\) and \(3\) (add to \(11\)).
Rewrite: \(4x^2 + 8x + 3x + 6\)
Group: \((4x^2 + 8x) + (3x + 6)\)
Factor: \(4x(x + 2) + 3(x + 2)=(4x + 3)(x + 2)\)
Overall: \(5(4x + 3)(x + 2)\)

Step15: Factor \(10x^2 - 32x + 24\) (GCF is \(2\))

Factor out \(2\): \(2(5x^2 - 16x + 12)\)
Factor \(5x^2 - 16x + 12\). Multiply \(5\times12 = 60\). Numbers \(-10\) and \(-6\) (add to \(-16\)).
Rewrite: \(5x^2 - 10x - 6x + 12\)
Group: \((5x^2 - 10x) + (-6x + 12)\)
Factor: \(5x(x - 2) - 6(x - 2)=(5x - 6)(x - 2)\)
Overall: \(2(5x - 6)(x - 2)\)

Answer:

1. \((2x + 1)(x + 2)\)
2. \((3n + 2)(n + 1)\)
3. \((2y - 1)(y + 5)\)
4. \((3g - 1)(g - 2)\)
5. \((2k - 5)(k - 3)\)
6. \((3p - 2)(3p + 4)\)
7. \((3m - 2)(2m - 3)\)
8. \((c - 2)(3c + 8)\)
9. \((2b + 3)^2\)
10. \(2(4x - 5)(x + 1)\)
11. \(3(2a - 3)(a + 3)\)
12. \(4(3x - 1)(5x + 2)\)
13. \(8(2x - 1)(x - 1)\)
14. \(5(4x + 3)(x + 2)\)
15. \(2(5x - 6)(x - 2)\)