To determine which option matches the polynomial [tex]\( x^4 - 5x^3 + 8x^2 - 5x - 1 \)[/tex], we will expand each given choice and compare the expanded form to the provided polynomial.
Let's start with option a):
[tex]\[ (x-1)^2(x^2 - 3x + 1) \][/tex]
First, expand [tex]\((x-1)^2\)[/tex]:
[tex]\[ (x-1)^2 = x^2 - 2x + 1 \][/tex]
Next, multiply this with [tex]\((x^2 - 3x + 1)\)[/tex]:
[tex]\[
(x^2 - 2x + 1)(x^2 - 3x + 1)
\][/tex]
First, distribute [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x^2 - 3x + 1) = x^4 - 3x^3 + x^2
\][/tex]
Next, distribute [tex]\(-2x\)[/tex]:
[tex]\[
-2x(x^2 - 3x + 1) = -2x^3 + 6x^2 - 2x
\][/tex]
Finally, distribute [tex]\(1\)[/tex]:
[tex]\[
1(x^2 - 3x + 1) = x^2 - 3x + 1
\][/tex]
Combine all these results:
[tex]\[
x^4 - 3x^3 + x^2 - 2x^3 + 6x^2 - 2x + x^2 - 3x + 1
\][/tex]
Simplify by combining like terms:
[tex]\[
x^4 - 5x^3 + 8x^2 - 5x + 1
\][/tex]
We see that the expanded polynomial matches the given polynomial, [tex]\( x^4 - 5x^3 + 8x^2 - 5x - 1 \)[/tex]. Thus, option a) is the correct answer.
To confirm, we can briefly check the other options without full expansion:
- Option b) would result in [tex]\((x+1)^2 = x^2 + 2x + 1\)[/tex], which creates positive [tex]\(x^3\)[/tex] terms and will not match the negative term [tex]\( -5x^3 \)[/tex] in the original polynomial.
- Option c) involves [tex]\((x-1)^2(x^2 + 3x + 1)\)[/tex], introducing positive [tex]\(3x\)[/tex] terms, which are incorrect.
- Option d) involves [tex]\((x-1)^2(x^2 - 3x - 1)\)[/tex], leading to an extra [tex]\(-x^2\)[/tex] term that does not match.
Thus, the correct answer is:
[tex]\[ \boxed{(x-1)^2(x^2 - 3x + 1)} \][/tex]
