Find all rational zeros of the polynomial Px 6x4 7x3 13x2

Find all rational zeros of the polynomial

P(x) = 6x⁴ 7x³ 13x² + 4x + 4

write the polynomial in factored form

Solution

Solution:

Given

P(x) = 6x⁴ 7x³ 13x³ + 4x + 4

As we can see that for x = -1

we get p(-1) = 0

so the x+1 is factor of P(x)

thus we can write

P(x)= (x+10) (6x³ -13x² +4)

As we again factorise

(6x³ -13x² +4)

for x = 2

we get

(6x³ -13x² +4) = 0

therefore x -2 is factor of (6x³ -13x² +4)

therefore (6x³ -13x² +4) = (x-2) (6x² -x -2)

again

(6x² -x -2) can be written as (2x+1) (3x-2)

So from above

P(x) = 6x⁴ 7x³ 13x³ + 4x + 4

can be written in factored form

P(x) = (x+1) (x-2) (2x+1) (3x-2)

Now

all rational zeros of the polynomials

are x = -1 , 2 , -1/2 and 2/3

Answer