"Find all rational zeros of the polynomial, and..." was answered by our community

EunoR

Answered question

2021-05-04

Find all rational zeros of the polynomial, and write the polynomial in factored form.

P (x) = 4 x^{4} - 37 x^{2} + 9

Answer & Explanation

komunidadO

Skilled2021-05-06Added 86 answers

Step 1
Given, $P (x) = 4 x^{4} - 37 x^{2} + 9$
For zeroes, substituting,P(x)=0
$\Rightarrow 4 x^{4} - 37 x^{2} + 9 = 0$
$\Rightarrow 4 x^{4} - 36 x^{2} - x^{2} + 9 = 0$
$\Rightarrow 4 x^{2} (x^{2} - 9) - (x^{2} - 9) = 0$
$\Rightarrow (x^{2} - 9) (4 x^{2} - 1) = 0$
$\Rightarrow (x^{2} - 3^{2}) ({(2 x)}^{2} - 1) = 0$
$\Rightarrow (x + 3) (x - 3) (2 x + 1) (2 x - 1) = 0$
$(U s i n g, (a^{2} - b^{2}) = (a + b) (a - b))$
$\Rightarrow (x + 3) = 0$ , or, (x-3)=0, or , (2x+1)=0, or, (2x-1)=0
$\Rightarrow x = - 3, 3, - \frac{1}{2}, \frac{1}{2}$
Hence, zeroes are: $x = (- 3, - \frac{1}{2}, \frac{1}{2}, 3)$ .
Step 2
Required factorized form is:
$4 x^{4} - 37 x^{2} + 9 = (x + 3) (x - 3) (2 x + 1) (2 x - 1)$

Jeffrey Jordon

Expert2022-08-02Added 2605 answers

Answer is given below (on video)

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