Studdy Solution
STEP 1
1. The polynomial f(x)=−4x3+13x2+13x−4 is given with its zeros.
2. The zeros of the polynomial are −1,41, and 4.
3. The polynomial can be expressed in factored form using its zeros.
STEP 2
1. Use the given zeros to write the polynomial in factored form.
2. Ensure the leading coefficient matches the original polynomial.
STEP 3
Write the polynomial in terms of its zeros. Since the zeros are −1,41, and 4, the factors corresponding to these zeros are (x+1),(x−41), and (x−4).
f(x)=a(x+1)(x−41)(x−4)
Where a is a constant to be determined.
STEP 4
Determine the value of a by expanding the factored form and matching it with the leading coefficient of the original polynomial.
First, multiply the factors:
(x−41)(x−4)=x2−4x−41x+1=x2−417x+1
Now multiply by (x+1):
(x+1)(x2−417x+1)=x3−417x2+x+x2−417x+1
Combine like terms:
=x3−413x2−413x+1
To match the leading coefficient of the original polynomial, set:
a⋅1=−4
Thus, a=−4.
STEP 5
Substitute a=−4 back into the factored form:
f(x)=−4(x+1)(x−41)(x−4)
Simplify the expression:
f(x)=−4(x+1)(x−41)(x−4)
=−4(x+1)(44x−1)(x−4)
=−(x+1)(4x−1)(x−4)
This is the complete factored form of f(x).
The complete factored form of f(x) is:
f(x)=−4(x+1)(x−41)(x−4)
