Let's factorize the polynomial [tex]\( P(x) = x^3 + 8x^2 + 17x + 10 \)[/tex] step-by-step to find its complete factorization.
1. Identify the polynomial:
[tex]\[
P(x) = x^3 + 8x^2 + 17x + 10
\][/tex]
2. Check for possible rational roots: To factorize this polynomial, we can use the Rational Root Theorem which suggests that any rational root [tex]\( p/q \)[/tex] of the polynomial, where [tex]\( p \)[/tex] is a factor of the constant term (10) and [tex]\( q \)[/tex] is a factor of the leading coefficient (1), must be a candidate. Therefore, we have potential roots [tex]\( \pm 1, \pm 2, \pm 5, \pm 10 \)[/tex].
3. Test the potential rational roots:
Evaluate the polynomial for these potential roots to see which one is actually a root of the polynomial.
When we test these values, we eventually find that:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
P(-1) = (-1)^3 + 8(-1)^2 + 17(-1) + 10 = -1 + 8 - 17 + 10 = 0
\][/tex]
Therefore, [tex]\( x = -1 \)[/tex] is a root, so [tex]\( (x + 1) \)[/tex] is a factor.
4. Perform synthetic division: Next, we do synthetic division of [tex]\( P(x) \)[/tex] by [tex]\( x + 1 \)[/tex].
[tex]\[
\begin{array}{r|rrrr}
-1 & 1 & 8 & 17 & 10 \\
& & -1 & -7 & -10 \\
\hline
& 1 & 7 & 10 & 0 \\
\end{array}
\][/tex]
The quotient is [tex]\( x^2 + 7x + 10 \)[/tex] with a remainder of 0, confirming that [tex]\( P(x) = (x + 1)(x^2 + 7x + 10) \)[/tex].
5. Factorize the quadratic: Now, we factorize the quadratic polynomial [tex]\( x^2 + 7x + 10 \)[/tex].
Find two numbers that multiply to 10 and add to 7. Those numbers are 2 and 5:
[tex]\[
x^2 + 7x + 10 = (x + 2)(x + 5)
\][/tex]
6. Combine the factors:
[tex]\[
P(x) = (x + 1)(x + 2)(x + 5)
\][/tex]
7. Verify the factorization: To ensure the factorization is correct, we can multiply the factors back together:
[tex]\[
(x + 1)(x + 2)(x + 5)
\][/tex]
Multiply [tex]\( (x + 1)(x + 2) \)[/tex]:
[tex]\[
(x + 1)(x + 2) = x^2 + 3x + 2
\][/tex]
Then multiply [tex]\( (x^2 + 3x + 2)(x + 5) \)[/tex]:
[tex]\[
(x^2 + 3x + 2)(x + 5) = x^3 + 5x^2 + 3x^2 + 15x + 2x + 10 = x^3 + 8x^2 + 17x + 10
\][/tex]
The polynomial matches the original, confirming the factorization.
Therefore, the complete factorization of the polynomial [tex]\( x^3 + 8x^2 + 17x + 10 \)[/tex] is:
[tex]\[
\boxed{(x + 1)(x + 2)(x + 5)}
\][/tex]
The correct answer is option C.
