Given the polynomial [tex]\( P(x) = x^3 - 7x^2 + 15x - 9 \)[/tex] and knowing that [tex]\( x - 1 \)[/tex] is a factor of this polynomial, we need to find the complete factorization.
First, let's use the factor theorem, which states that if [tex]\( x - c \)[/tex] is a factor of the polynomial [tex]\( P(x) \)[/tex], then [tex]\( P(c) = 0 \)[/tex].
We are given that [tex]\( x - 1 \)[/tex] is a factor, so [tex]\( P(1) = 0 \)[/tex]. This confirms that [tex]\( x = 1 \)[/tex] is a root of the polynomial.
Since [tex]\( P(x) \)[/tex] is a cubic polynomial ([tex]\( x^3 \)[/tex]), it can be factorized into the form:
[tex]\[ P(x) = (x - r_1)(x - r_2)(x - r_3) \][/tex]
Given that [tex]\( x - 1 \)[/tex] is a factor, we can rewrite:
[tex]\[ P(x) = (x - 1)(Q(x)) \][/tex]
where [tex]\( Q(x) \)[/tex] is a quadratic polynomial.
However, we already have the factorization result:
[tex]\[ P(x) = (x - 3)^2 (x - 1) \][/tex]
Breaking down the solution:
- The root [tex]\( x - 1 \)[/tex] corresponds directly to the factor [tex]\( x - 1 \)[/tex].
- The term [tex]\( (x - 3)^2 \)[/tex] indicates that [tex]\( x - 3 \)[/tex] is a factor with multiplicity 2.
Thus, the complete factorization of [tex]\( P(x) \)[/tex] is:
[tex]\[ P(x) = (x - 3)(x - 3)(x - 1) \][/tex]
Comparing with the given options, the correct answer is:
C. [tex]\((x - 3)(x - 3)(x - 1)\)[/tex]
