let px 4X3 X2 32x 8 a Find p3 using the Remainder Theorem

let p(x) 4X^3 + X^2 - 32x - 8. (a) Find p(3) using the Remainder Theorem. P(3) = (b) Given that c = -1/4 is a zero of p, use this to find the remaining zeros of p. (Enter your answers as a comma-separated list. If there are no other zeros, enter the zero given.) x =

Solution

we have p(x)=4x³+x²-32x-8

a) hence p(3) can be obtained by replacing x=3 in the equation

hence p(3)=4*(3)³+3²-32*3-8=108+9-96-8=13 [answer]

b) c=-1/4=-0.25 is a zero of p

that means (x+0.25) is a factor of p

so

4x³+x²-32x-8

= 4x²(x+0.25)+0(x+0.25)-32(x+0.25)

=4x²(x+0.25)-32(x+0.25)

=(x+0.25)(4x²-32)

hence p(x)=0

implies (x+0.25)(4x²-32)=0

or, x+0.25=0....(i)              or 4x²-32=0...(ii)

from (i) we get x=-0.25 is a zero of p

from (ii) we get 4x²=32 or, x²=8 or x=sqrt(8)=+2.828427125 or -2.828427125

hence the other zeroes of p are

+2.828427125,-2.828427125 [answer]