Expert Answer to "a) Find the rational zeros and then the..."

College
Calculus and Analysis
Integral Calculus
Rational exponents and radicals

Zoe Oneal

Answered question

2021-01-02

a) Find the rational zeros and then the other zeros of the polynomial function $(x) = x^{3} - 4 x^{2} + 2 x + 4, that is, solve f (x) = 0 .$
b) Factor $f (x)$ into linear factors.

Answer & Explanation

opsadnojD

Skilled2021-01-03Added 95 answers

Step 1
a) We graph $f (x) = x^{3} - 4 x^{2} + 2 x + 4$ using a graphing calculator to find the rational zeros.

It cuts the graph at $x = 2$
Means $x = 2$ is a zero.
Step 2
Then we use $x = 2$ to find the other factor. We use synthetic division.

Another factor is $x^{2} - 2 x - 2 .$
Step 3
Then we use the quadratic formula to find the zeros fromit
$x^{2} - 2 x - 2 = 0$
$x = \frac{2 \pm \sqrt{{(- 2)}^{2} - 4 (1) (- 2)}}{2 (1)}$
$x = \frac{2 \pm \sqrt{12}}{2}$
$x = \frac{2 \pm 2 \sqrt{3}}{2}$
$x = 1 \pm \sqrt{3}$
Answer(a): $1 - \sqrt{3}, 2, 1 \pm \sqrt{3}$
Step 4
b) Linear factored form of f is:
$f (x) = [x - (1 - \sqrt{3})] (x - 2) [x - (1 + \sqrt{3})]$
$f (x) = (x - 1 + \sqrt{3}) (x - 2) (x - 1 - \sqrt{3})$
Answer: $(x - 1 + \sqrt{3}) (x - 2) (x - 1 - \sqrt{3})$

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