Sure! Let's analyze the given polynomials and match them with their factored forms step-by-step.
1. Polynomial: [tex]\( 8x^3 + 16x^2 \)[/tex]
We need to factor this polynomial. Notice that [tex]\( 8x^3 + 16x^2 \)[/tex] can be factored by taking out the common factor:
[tex]\[
8x^3 + 16x^2 = 8x^2(x + 2)
\][/tex]
So the factored form is [tex]\( 8x^2(x + 2) \)[/tex].
2. Polynomial: [tex]\( (2x - 4)(x + 1) \)[/tex]
Let's expand [tex]\( (2x - 4)(x + 1) \)[/tex]:
[tex]\[
(2x - 4)(x + 1) = 2x(x + 1) - 4(x + 1) = 2x^2 + 2x - 4x - 4 = 2x^2 - 2x - 4
\][/tex]
None of the given expanded forms match this polynomial exactly. Hence, it's unnecessary in our final match.
3. Polynomial: [tex]\( 2x^2 - x + 8x - 4 \)[/tex]
Let's combine the like terms in the polynomial:
[tex]\[
2x^2 - x + 8x - 4 = 2x^2 + 7x - 4
\][/tex]
We need to match this to a potential factored form. One candidate is [tex]\( (2x - 1)(x + 4) \)[/tex]. Let's expand it to verify:
[tex]\[
(2x - 1)(x + 4) = 2x(x + 4) - 1(x + 4) = 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4
\][/tex]
It matches exactly, so the factored form of [tex]\( 2x^2 + 7x - 4 \)[/tex] is [tex]\( (2x - 1)(x + 4) \)[/tex].
4. Polynomial: [tex]\( \left(8x^2 + 1\right)(x + 2) \)[/tex]
We don’t have an expanded polynomial from our set that matches this form after factoring since our previous checks don’t show a match.
So the solutions are:
- The polynomial [tex]\( 8x^3 + 16x^2 \)[/tex] matches with [tex]\( 8x^2(x + 2) \)[/tex].
- The polynomial [tex]\( 2x^2 - x + 8x - 4 \)[/tex] matches with [tex]\( (2x - 1)(x + 4) \)[/tex].
Final matches are:
1. [tex]\( 8x^3 + 16x^2 \leftrightarrow 8x^2(x + 2) \)[/tex]
2. [tex]\( 2x^2 - x + 8x - 4 \leftrightarrow (2x - 1)(x + 4) \)[/tex]
