Factoring by Grouping Worksheet - Math Monks - Free Printable

Final Answer:
1. $(x - 2)(x^2 + 5)$
2. $(6x + 5)(2x^2 + 3)$
3. $(2 + 5x^5)(x^2 + 3x^3)$ or better: $(x^2 + 3x^3)(2 + 5x^5)$ — but standard grouping gives: $(2x^2 + 6x^3) + (5x^7 + 15x^8) = 2x^2(1 + 3x) + 5x^7(1 + 3x) = (1 + 3x)(2x^2 + 5x^7) = x^2(1 + 3x)(2 + 5x^5)$
Actually, let’s do each cleanly:

1. $x^3 - 2x^2 + 5x - 10 = (x^3 - 2x^2) + (5x - 10) = x^2(x - 2) + 5(x - 2) = (x - 2)(x^2 + 5)$
2. $12x^3 + 18x^2 + 10x + 15 = (12x^3 + 18x^2) + (10x + 15) = 6x^2(2x + 3) + 5(2x + 3) = (2x + 3)(6x^2 + 5)$
3. $2x^2 + 6x^3 + 5x^7 + 15x^8 = (2x^2 + 6x^3) + (5x^7 + 15x^8) = 2x^2(1 + 3x) + 5x^7(1 + 3x) = (1 + 3x)(2x^2 + 5x^7) = x^2(1 + 3x)(2 + 5x^5)$
But usually we factor out common powers: final answer: $x^2(1 + 3x)(2 + 5x^5)$
4. $6x^4 - 24x^3 - 5x + 20 = (6x^4 - 24x^3) + (-5x + 20) = 6x^3(x - 4) -5(x - 4) = (x - 4)(6x^3 - 5)$
5. $4a^2 - 21b^3 + 6ab - 14ab^2$ → group as $(4a^2 + 6ab) + (-21b^3 - 14ab^2) = 2a(2a + 3b) -7b^2(3b + 2a) = (2a + 3b)(2a - 7b^2)$
6. $10x^2 + 5x + 4x + 2 = (10x^2 + 5x) + (4x + 2) = 5x(2x + 1) + 2(2x + 1) = (2x + 1)(5x + 2)$
7. $140xy - 60x^2 + 168y - 72x = (-60x^2 + 140xy) + (-72x + 168y) = -20x(3x - 7y) -24(3x - 7y) = (3x - 7y)(-20x - 24) = -4(3x - 7y)(5x + 6)$
Or better: factor out -4: $= -4(5x + 6)(3x - 7y)$ — but often written as $(7y - 3x)(20x + 24)$, but simplest integer factors: $-4(5x + 6)(3x - 7y)$
However, many teachers prefer positive leading term: rewrite as $(7y - 3x)(20x + 24) = 4(7y - 3x)(5x + 6)$
Let’s check: $4(7y - 3x)(5x + 6) = 4[35xy + 42y -15x^2 -18x] = 140xy + 168y -60x^2 -72x$ ✓
So: $4(7y - 3x)(5x + 6)$
8. $21xy - 12b^2 + 14xb - 18by = (21xy + 14xb) + (-12b^2 - 18by) = 7x(3y + 2b) -6b(2b + 3y) = (3y + 2b)(7x - 6b)$

So final answers:

1. $(x - 2)(x^2 + 5)$
2. $(2x + 3)(6x^2 + 5)$
3. $x^2(1 + 3x)(2 + 5x^5)$
4. $(x - 4)(6x^3 - 5)$
5. $(2a + 3b)(2a - 7b^2)$
6. $(2x + 1)(5x + 2)$
7. $4(7y - 3x)(5x + 6)$
8. $(3y + 2b)(7x - 6b)$