To factor the polynomial [tex]\( f(x) = 3x^3 + 17x^2 - 54x + 16 \)[/tex] completely, given that [tex]\( k = 2 \)[/tex] is a zero of [tex]\( f(x) \)[/tex], we can follow these steps:
1. Verify that [tex]\( x = 2 \)[/tex] is a root: Given [tex]\( k = 2 \)[/tex] is a zero of [tex]\( f(x) \)[/tex], [tex]\( f(2) = 0 \)[/tex]. Therefore, [tex]\( x - 2 \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
2. Divide [tex]\( f(x) \)[/tex] by [tex]\( x - 2 \)[/tex]: To factor [tex]\( f(x) \)[/tex] further, we divide [tex]\( f(x) \)[/tex] by [tex]\( x - 2 \)[/tex] using polynomial division or synthetic division. This will give us a quotient and possibly a remainder. Since [tex]\( x = 2 \)[/tex] is a root, the remainder will be zero.
Performing synthetic division or polynomial division:
- The quotient obtained is [tex]\( 3x^2 + 23x - 8 \)[/tex].
3. Factor the quotient: Next, we need to factor the quotient [tex]\( 3x^2 + 23x - 8 \)[/tex]. To do this, we can look for factor pairs of the product of the leading coefficient (3) and the constant term (-8):
- Factor pair that sums to the middle coefficient (23) is found, which splits the middle term into two factors.
- Rewriting and factoring the quadratic:
[tex]\[
3x^2 + 23x - 8 = (x + 8)(3x - 1)
\][/tex]
4. Combine the factors: Now, we combine [tex]\( x - 2 \)[/tex] with the factors obtained from the quotient:
- The complete factorization of [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = (x - 2)(x + 8)(3x - 1)
\][/tex]
Therefore, the polynomial [tex]\( f(x) = 3x^3 + 17x^2 - 54x + 16 \)[/tex] can be factored completely as:
[tex]\[
f(x) = (x - 2)(x + 8)(3x - 1)
\][/tex]
