Question

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using

a graphing utility, use it to graph the function and verify the real zeros and the given function value.
n=3;
-2 and 7+5 i are zeros;
f(2)=200
f(x) =
(Type an expression using x as the variable. Simplify your answer.)
possible
4

Answer 1

f(x) = x³ -12x² +46x +148

Explanation:

When p is a root of polynomial function f(x), (x -p) is a factor. When the coefficients are real, any complex roots come in conjugate pairs.

Factored form

Given the two roots of f(x), we know the third root is the conjugate of the given complex root. The factored form will be ...

f(x) = (x -(-2))(x -(7 +5i))(x -(7 -5i))

Rearranging a bit, this is ...

f(x) = (x +2)((x -7) -5i)((x -7) +5i)

The latter two factors are recognizable as the factors of the difference of squares, so this is ...

f(x) = (x +2)((x -7)² -(5i)²) = (x +2)((x -7)² +25)

Standard form

Multiplying the factors, we have ...

f(x) = (x +2)(x² -14x +49 +25) = (x +2)(x² -14x +74)

f(x) = x³ -14x² +74x +2x² -28x +148 . . . . . use the distributive property

f(x) = x³ -12x² +46x +148 . . . . . . . . . . collect terms