Answer
Here are the factorizations for the given cubic polynomials:
(a) \( x^{3}+4 x^{2}+x-6 = (x-1)(x+2)(x+3) \)
(b) \( x^{3}-3 x^{2}-4 x+12 = (x-3)(x-2)(x+2) \)
(c) \( x^{3}+8 x^{2}-11 x-18 = (x-2)(x+1)(x+9) \)
(d) \( 2 x^{3}-3 x^{2}-2 x+3 = (x-1)(x+1)(2x-3) \)
(e) \( 4 x^{3}-5 x^{2}-23 x+6 = (x-3)(x+2)(4x-1) \)
(f) \( 2 x^{3}-16 x^{2}+10 x+28 = 2(x-7)(x-2)(x+1) \)
(g) \( x^{3}-7 x-6 = (x-3)(x+1)(x+2) \)
(h) \( x^{3}-13 x-12 = (x-4)(x+1)(x+3) \)
(i) \( 2 x^{3}+7 x^{2}+4 x-4 = (x+2)^{2}(2x-1) \)
(j) \( -4 x^{3}+3 x+1 = -(x-1)(2x+1)^{2} \)
Solution
Factor the expression by following steps:
- step0: Factor:
\(-4x^{3}+3x+1\)
- step1: Calculate:
\(-4x^{3}-4x^{2}-x+4x^{2}+4x+1\)
- step2: Rewrite the expression:
\(-x\times 4x^{2}-x\times 4x-x+4x^{2}+4x+1\)
- step3: Factor the expression:
\(-x\left(4x^{2}+4x+1\right)+4x^{2}+4x+1\)
- step4: Factor the expression:
\(\left(-x+1\right)\left(4x^{2}+4x+1\right)\)
- step5: Factor the expression:
\(\left(-x+1\right)\left(2x+1\right)^{2}\)
- step6: Factor the expression:
\(-\left(x-1\right)\left(2x+1\right)^{2}\)
Factor the expression \( 4 x^{3}-5 x^{2}-23 x+6 \).
Factor the expression by following steps:
- step0: Factor:
\(4x^{3}-5x^{2}-23x+6\)
- step1: Calculate:
\(4x^{3}+7x^{2}-2x-12x^{2}-21x+6\)
- step2: Rewrite the expression:
\(x\times 4x^{2}+x\times 7x-x\times 2-3\times 4x^{2}-3\times 7x+3\times 2\)
- step3: Factor the expression:
\(x\left(4x^{2}+7x-2\right)-3\left(4x^{2}+7x-2\right)\)
- step4: Factor the expression:
\(\left(x-3\right)\left(4x^{2}+7x-2\right)\)
- step5: Factor the expression:
\(\left(x-3\right)\left(x+2\right)\left(4x-1\right)\)
Factor the expression \( x^{3}-3 x^{2}-4 x+12 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{3}-3x^{2}-4x+12\)
- step1: Calculate:
\(x^{3}-4x-3x^{2}+12\)
- step2: Rewrite the expression:
\(x\times x^{2}-x\times 4-3x^{2}+3\times 4\)
- step3: Factor the expression:
\(x\left(x^{2}-4\right)-3\left(x^{2}-4\right)\)
- step4: Factor the expression:
\(\left(x-3\right)\left(x^{2}-4\right)\)
- step5: Factor the expression:
\(\left(x-3\right)\left(x-2\right)\left(x+2\right)\)
Factor the expression \( x^{3}-7 x-6 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{3}-7x-6\)
- step1: Calculate:
\(x^{3}+3x^{2}+2x-3x^{2}-9x-6\)
- step2: Rewrite the expression:
\(x\times x^{2}+x\times 3x+x\times 2-3x^{2}-3\times 3x-3\times 2\)
- step3: Factor the expression:
\(x\left(x^{2}+3x+2\right)-3\left(x^{2}+3x+2\right)\)
- step4: Factor the expression:
\(\left(x-3\right)\left(x^{2}+3x+2\right)\)
- step5: Factor the expression:
\(\left(x-3\right)\left(x+1\right)\left(x+2\right)\)
Factor the expression \( 2 x^{3}-16 x^{2}+10 x+28 \).
Factor the expression by following steps:
- step0: Factor:
\(2x^{3}-16x^{2}+10x+28\)
- step1: Rewrite the expression:
\(2x^{3}-2\times 8x^{2}+2\times 5x+2\times 14\)
- step2: Factor the expression:
\(2\left(x^{3}-8x^{2}+5x+14\right)\)
- step3: Factor the expression:
\(2\left(x-7\right)\left(x^{2}-x-2\right)\)
- step4: Factor the expression:
\(2\left(x-7\right)\left(x-2\right)\left(x+1\right)\)
Factor the expression \( x^{3}+8 x^{2}-11 x-18 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{3}+8x^{2}-11x-18\)
- step1: Calculate:
\(x^{3}+10x^{2}+9x-2x^{2}-20x-18\)
- step2: Rewrite the expression:
\(x\times x^{2}+x\times 10x+x\times 9-2x^{2}-2\times 10x-2\times 9\)
- step3: Factor the expression:
\(x\left(x^{2}+10x+9\right)-2\left(x^{2}+10x+9\right)\)
- step4: Factor the expression:
\(\left(x-2\right)\left(x^{2}+10x+9\right)\)
- step5: Factor the expression:
\(\left(x-2\right)\left(x+1\right)\left(x+9\right)\)
Factor the expression \( x^{3}-13 x-12 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{3}-13x-12\)
- step1: Calculate:
\(x^{3}+4x^{2}+3x-4x^{2}-16x-12\)
- step2: Rewrite the expression:
\(x\times x^{2}+x\times 4x+x\times 3-4x^{2}-4\times 4x-4\times 3\)
- step3: Factor the expression:
\(x\left(x^{2}+4x+3\right)-4\left(x^{2}+4x+3\right)\)
- step4: Factor the expression:
\(\left(x-4\right)\left(x^{2}+4x+3\right)\)
- step5: Factor the expression:
\(\left(x-4\right)\left(x+1\right)\left(x+3\right)\)
Factor the expression \( x^{3}+4 x^{2}+x-6 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{3}+4x^{2}+x-6\)
- step1: Calculate:
\(x^{3}+5x^{2}+6x-x^{2}-5x-6\)
- step2: Rewrite the expression:
\(x\times x^{2}+x\times 5x+x\times 6-x^{2}-5x-6\)
- step3: Factor the expression:
\(x\left(x^{2}+5x+6\right)-\left(x^{2}+5x+6\right)\)
- step4: Factor the expression:
\(\left(x-1\right)\left(x^{2}+5x+6\right)\)
- step5: Factor the expression:
\(\left(x-1\right)\left(x+2\right)\left(x+3\right)\)
Factor the expression \( 2 x^{3}+7 x^{2}+4 x-4 \).
Factor the expression by following steps:
- step0: Factor:
\(2x^{3}+7x^{2}+4x-4\)
- step1: Calculate:
\(2x^{3}-x^{2}+8x^{2}-4x+8x-4\)
- step2: Rewrite the expression:
\(x^{2}\times 2x-x^{2}+4x\times 2x-4x+4\times 2x-4\)
- step3: Factor the expression:
\(x^{2}\left(2x-1\right)+4x\left(2x-1\right)+4\left(2x-1\right)\)
- step4: Factor the expression:
\(\left(x^{2}+4x+4\right)\left(2x-1\right)\)
- step5: Factor the expression:
\(\left(x+2\right)^{2}\left(2x-1\right)\)
Factor the expression \( 2 x^{3}-3 x^{2}-2 x+3 \).
Factor the expression by following steps:
- step0: Factor:
\(2x^{3}-3x^{2}-2x+3\)
- step1: Rewrite the expression:
\(x^{2}\times 2x-x^{2}\times 3-2x+3\)
- step2: Factor the expression:
\(x^{2}\left(2x-3\right)-\left(2x-3\right)\)
- step3: Factor the expression:
\(\left(x^{2}-1\right)\left(2x-3\right)\)
- step4: Factor the expression:
\(\left(x-1\right)\left(x+1\right)\left(2x-3\right)\)
Here are the factorizations for the given cubic polynomials:
(a) \( x^{3}+4 x^{2}+x-6 \)
**Factorization:** \( (x-1)(x+2)(x+3) \)
(b) \( x^{3}-3 x^{2}-4 x+12 \)
**Factorization:** \( (x-3)(x-2)(x+2) \)
(c) \( x^{3}+8 x^{2}-11 x-18 \)
**Factorization:** \( (x-2)(x+1)(x+9) \)
(d) \( 2 x^{3}-3 x^{2}-2 x+3 \)
**Factorization:** \( (x-1)(x+1)(2x-3) \)
(e) \( 4 x^{3}-5 x^{2}-23 x+6 \)
**Factorization:** \( (x-3)(x+2)(4x-1) \)
(f) \( 2 x^{3}-16 x^{2}+10 x+28 \)
**Factorization:** \( 2(x-7)(x-2)(x+1) \)
(g) \( x^{3}-7 x-6 \)
**Factorization:** \( (x-3)(x+1)(x+2) \)
(h) \( x^{3}-13 x-12 \)
**Factorization:** \( (x-4)(x+1)(x+3) \)
(i) \( 2 x^{3}+7 x^{2}+4 x-4 \)
**Factorization:** \( (x+2)^{2}(2x-1) \)
(j) \( -4 x^{3}+3 x+1 \)
**Factorization:** \( -(x-1)(2x+1)^{2} \)
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