Question

26 fill in the blank 1 point
factor completely. then solve.
$6x^3 - 150x$
factored form is (enter the answer with no spaces)
the solutions, from smallest to largest, are,, and.
27 multiple choice 1 point
factor completely.
$x^4 - 6x^2 - 27$
$\big(x^2 - 3\big)\big(x^2 + 9\big)$
$\big(x^2 - 3\big)\big(x + 3\big)\big(x - 3\big)$
$\big(x + 3\big)\big(x - 9\big)$
$\big(x^2 + 3\big)\big(x + 3\big)\big(x - 3\big)$
clear my selection
28 fill in the blank 1 point
find all real solutions. $x^3 + 6x^2 + 3x - 10 = 0$
the solutions, in order from smallest to greatest, are,, and.

Explanation:

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Question 26

Step1: Factor out GCF

Factor $6x$ from $6x^3-150x$:
$6x(x^2 - 25)$

Step2: Factor difference of squares

Rewrite $x^2-25$ as $x^2-5^2$, then factor:
$6x(x-5)(x+5)$

Step3: Set factors to 0

Solve $6x=0$, $x-5=0$, $x+5=0$:
$x=0$, $x=5$, $x=-5$

Step4: Order solutions

Sort solutions from smallest to largest:
$-5, 0, 5$

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Question 27

Step1: Substitute $u=x^2$

Rewrite $x^4-6x^2-27$ as $u^2-6u-27$

Step2: Factor quadratic in $u$

Find two terms that multiply to $-27$ and add to $-6$:
$(u-9)(u+3)$

Step3: Substitute back $u=x^2$

$(x^2-9)(x^2+3)$

Step4: Factor difference of squares

Rewrite $x^2-9$ as $x^2-3^2$, then factor:
$(x+3)(x-3)(x^2+3)$

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Question 28

Step1: Test rational roots

Use Rational Root Theorem, test $x=1$:
$1^3+6(1)^2+3(1)-10=1+6+3-10=0$

Step2: Factor out $(x-1)$

Use polynomial division or synthetic division:
$x^3+6x^2+3x-10=(x-1)(x^2+7x+10)$

Step3: Factor quadratic

Factor $x^2+7x+10$:
$(x+2)(x+5)$

Step4: Set factors to 0

Solve $x-1=0$, $x+2=0$, $x+5=0$:
$x=1$, $x=-2$, $x=-5$

Step5: Order solutions

Sort solutions from smallest to largest:
$-5, -2, 1$

Answer:

Question 26:

Factored form: $6x(x-5)(x+5)$
Solutions: $-5$, $0$, $5$

Question 27:

D. $(x^2 + 3)(x + 3)(x - 3)$

Question 28:

Solutions: $-5$, $-2$, $1$