Solved: "Find all rational zeros of the polynomial, and..."

chillywilly12a

Answered question

2021-04-23

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

P (x) = 12 x^{3} - 25 x^{2} + x + 2

Bella

Skilled2021-04-25Added 81 answers

Step 1
Consider the given function as

P (x) = 12 x^{3} - 25 x^{2} + x + 2

Step 2
Let x =2 is a zero

P (2) = 12 {(2)}^{3} - 25 {(2)}^{2} + 2 + 2

=12(8)-25(4)+4
=96-100+4
=0
So, x=2 is a zero.
Now find the other zeros as
Now

\frac{12 x^{3} - 25 x^{2} + x + 2}{x - 2} = \frac{(3 x - 1) (4 x^{2} - 7 x - 2)}{x - 2}

= \frac{(3 x - 1) [4 x^{2} - (8 - 1) x - 2]}{x - 2}

= \frac{(3 x - 1) [4 x^{2} - 8 x + 1 x - 2]}{x - 2}

= \frac{(3 x - 1) [4 x (x - 2) + 1 (x - 2)]}{x - 2}

= \frac{(3 x - 1) (x - 2) (4 x + 1)}{x - 2}

=(3x-1)(4x+1)
Step 3
Hence, the zero are x = 2,

\frac{1}{3}, \frac{- 1}{4}

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