(a) find the rational zeros and then the other zeros of the polynomial function (x)=x3-4x2+2x+4, that is, solve f(x)=0. (type an exact answer, using r

Dottie Parra 2020-12-24 Answered

(a) find the rational zeros and then the other zeros of the polynomial function \((x)=x3-4x^2+2x+4\), that is, solve
\(f(x)=0.\) (type an exact answer, using radicals as needed. Simplify your answer. Use a comma to separate answers as needed.)
(b) Factor f(x) into linear factors. (type an exact answer, using radicals as needed. Simplify your answer. Type an expression using x as the variable.)

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Expert Answer

SoosteethicU
Answered 2020-12-25 Author has 18193 answers

a) We graph \(f(x) = x^{3} - 4x^{2} + 2x + 4\) using a graphing calculator to find the rational zeros.
image
It cuts the graph at \(x = 2\)
Means \(x = 2\) is a zero.
Then we use \(x=2\) to find the other factor. We use synthetic division.
Another factor is \(x^{2}-2x-2.\)
Then we use the quadratic formula to find the zeros from it
\(x^{2} - 2x - 2 = 0\)
\(x=\frac{2 \pm\sqrt{(-2)^{2}-4(1)(-2)}}{2(1)}\)
\(x=\frac{2 \pm \sqrt{12}}{2}\)
\(x=\frac{2 \pm 2 \sqrt{3}}{2}\)
\(x=1 \pm \sqrt{3}\)
Answer \((a): 1 -\sqrt{3},2,1+\sqrt{3}\)
image
b) Linear factored form of f is:
image
Answer: \((x-1+\sqrt{3})(x-2)(x-1-\sqrt{3})\)

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