Question

For which values of a and b, the zeros of q(x)=x^3+2x^2+a are also the zeros of the polynomial p(x)=x^5-x^4-4x^3+3x^2+3x+b?Which zeros of p(x) are not the zeros of q(x)?

Answer 1

Hello , here..☺★

______________solution_______________

Here, q (x) = x^3+2x^2+a
and P(x) = x^5 - x^4-4x^3+3x^2+3x+b
so, q(x) is a factor of p (x)
we use a division Algorithm...

★see the upper pic.★

if (x^3+2x^2+a) is a factor of (x^5-x^4-4x^3+3x^2+3x+b), than remainder should be zero.

=> -(1-a)x^2+(3+3a)x+(b-2a)=0
=>0.x^2+0.x=0
=>a+1=0
a = 0-1=-1
so the value of a is -1

and
=> b-2a=0
=> b= 2(-1)=0
the value of B is -2

for a = -1 and B = -2 , the zero of q(x) are also the zero of the polynomial P (x).

=> q (x ) = x^3+2x^2-1
P(x) = x^5-x^4-4x^3+3x^2+3x-2

we use the Euclid's division Algorithm.
DIVIDEND= DIVISOR × QUOTIENT+ REMAINDER

=> P(X) = (X^3+2X^2-1)(X^2-3X+2)+0
=> (X^3+2X^2-1) {X^2-2X-X+2}
=>(x^3+2x^2-1)(x-2)(x-1)

Therefore, The zero of P(x) is 2.
which is not the zero of q(x).

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Hope this helps you.
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