Question

factor completely. $5x^3 - 8x^2 + 15x - 24$
fill in the values of the letters in the answer: $(ax^2 + b)(cx + d)$
a= type your answer...
b= type your answer...
c= type your answer...
d= type your answer...
35 fill in the blank 1 point
solve. $x^3 + 5x^2 = -4x$
the solutions from smallest to largest are type your answer..., type your answer..., and type your answer...
36 fill in the blank 1 point
factor completely.
$27x^3 - 125$
fill in the values of a, b, c, d, and e in the final answer: $(ax + b)(cx^2 + dx + e)$
a= type your answer...

Explanation:

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Problem 1: Factor $5x^3 - 8x^2 + 15x - 24$

Step1: Group terms by pairs

$(5x^3 - 8x^2) + (15x - 24)$

Step2: Factor out GCF from each group

$x^2(5x - 8) + 3(5x - 8)$

Step3: Factor out common binomial

$(5x^2 + 15)(x - 8)$
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Problem 2: Solve $x^3 + 5x^2 = -4x$

Step1: Move all terms to left side

$x^3 + 5x^2 + 4x = 0$

Step2: Factor out GCF $x$

$x(x^2 + 5x + 4) = 0$

Step3: Factor quadratic trinomial

$x(x + 1)(x + 4) = 0$

Step4: Set each factor to 0

$x=0$, $x+1=0 \implies x=-1$, $x+4=0 \implies x=-4$

Step5: Order solutions smallest to largest

$-4, -1, 0$
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Problem 3: Factor $27x^3 - 125$

Step1: Recognize difference of cubes

$27x^3=(3x)^3$, $125=5^3$

Step2: Apply difference of cubes formula

$a^3-b^3=(a-b)(a^2+ab+b^2)$

Step3: Substitute $a=3x$, $b=5$

$(3x - 5)((3x)^2 + (3x)(5) + 5^2)=(3x-5)(9x^2+15x+25)$

Answer:

1. For $5x^3 - 8x^2 + 15x - 24$:

A=5, B=15, C=1, D=-8

1. For $x^3 + 5x^2 = -4x$:

-4, -1, 0

1. For $27x^3 - 125$:

A=3, B=-5, C=9, D=15, E=25