Let's match each polynomial with its factored form. Here, we know that each polynomial provided can be expanded and matched to one of the factored forms given.
We have the polynomial pairs to match:
1. [tex]\( x^3 - x^2 + 3x - 3 \)[/tex]
2. [tex]\( 2x^3 + 2x^2 - 12x \)[/tex]
3. [tex]\( x^2 - 6x + 8 \)[/tex]
We also have three factored forms:
1. [tex]\( 2x(x+3)(x-2) \)[/tex]
2. [tex]\( (x-4)(x-2) \)[/tex]
3. [tex]\( (x^2+3)(x-1) \)[/tex]
### Step-by-Step Matching
1. Match [tex]\( x^2 - 6x + 8 \)[/tex]:
The polynomial [tex]\( x^2 - 6x + 8 \)[/tex] can be matched by factoring:
[tex]\[ x^2 - 6x + 8 = (x-4)(x-2) \][/tex]
Thus:
[tex]\[ x^2 - 6x + 8 \quad \longrightarrow \quad (x-4)(x-2) \][/tex]
2. Match [tex]\( 2x^3 + 2x^2 - 12x \)[/tex]:
The polynomial [tex]\( 2x^3 + 2x^2 - 12x \)[/tex] can be factored:
[tex]\[ 2x^3 + 2x^2 - 12x = 2x(x+3)(x-2) \][/tex]
Thus:
[tex]\[ 2x^3 + 2x^2 - 12x \quad \longrightarrow \quad 2x(x+3)(x-2) \][/tex]
3. Match [tex]\( x^3 - x^2 + 3x - 3 \)[/tex]:
The polynomial [tex]\( x^3 - x^2 + 3x - 3 \)[/tex] can be factored:
[tex]\[ x^3 - x^2 + 3x - 3 = (x^2 + 3)(x-1) \][/tex]
Thus:
[tex]\[ x^3 - x^2 + 3x - 3 \quad \longrightarrow \quad (x^2 + 3)(x-1) \][/tex]
So, summarizing the matches:
- [tex]\( x^2 - 6x + 8 \quad \longrightarrow \quad (x-4)(x-2) \)[/tex]
- [tex]\( 2x^3 + 2x^2 - 12x \quad \longrightarrow \quad 2x(x+3)(x-2) \)[/tex]
- [tex]\( x^3 - x^2 + 3x - 3 \quad \longrightarrow \quad (x^2 + 3)(x-1) \)[/tex]
Thus, the final matched pairs are:
[tex]\[
\begin{aligned}
&\text{Factor:} \quad (x-4)(x-2) \quad &\text{Polynomial:} \quad x^2 - 6x + 8 \\
&\text{Factor:} \quad 2x(x+3)(x-2) \quad &\text{Polynomial:} \quad 2x^3 + 2x^2 - 12x \\
&\text{Factor:} \quad (x^2 + 3)(x-1) \quad &\text{Polynomial:} \quad x^3 - x^2 + 3x - 3 \\
\end{aligned}
\][/tex]
