Question

33 multiple choice 1 point kim, kathy, and amy are arguing over how to factor this polynomial. $x^3 + 1$ kim says factored form is $(x + 1)(x + 1)(x + 1)$. kathy says factored form is $(x + 1)(x^2 - x + 1)$ amy says factored form is $(x + 1)(x^2 + x + 1)$ who is correct? \bigcirc amy \bigcirc all of them \bigcirc kim \bigcirc kathy \bigcirc none of them clear my selection

Explanation:

First Problem (Factoring $x^6 - 14x^3 + 45$)

Step1: Recognize quadratic form

Let $u = x^3$, rewrite as $u^2 -14u +45$

Step2: Find pair for trinomial

Find two numbers: $a+b=-14$, $a \times b=45$. The numbers are $-9$ and $-5$.

Step3: Factor the quadratic

$(u-9)(u-5)$

Step4: Substitute back $u=x^3$

$(x^3-9)(x^3-5)$, so each $x$ has exponent $3$.

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Second Problem (Factoring $x^3 + 1$)

Step1: Use sum of cubes formula

Sum of cubes: $a^3+b^3=(a+b)(a^2-ab+b^2)$

Step2: Apply to $x^3+1$

Here $a=x$, $b=1$. So:
$$x^3+1=(x+1)(x^2 - x +1)$$
This matches Kathy's answer.

Answer:

Type of factoring: Substitution (quadratic in form); Add to: $-14$; Multiply to: $45$; Smaller number: $-9$; Larger number: $-5$; Exponent of $x$: $3$