Question

For x2 + 2x + 5 to be a factor of x4 + px2 + q, the values of p and q must be ____.

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Answer 1

Step-by-step explanation:

x^4+px^2+q.

=x^2(x^2+2x+5)-2x^3–5x^2+px^2+q.

=x^2(x^2+2x+5)-2x(x^2+2x+5)+4x^2+10x-5x^2+px^2+q.

=x^2(x^2+2x+5)-2x(x^2+2x+5)+(p-1). x^2+10x+q.

=x^2(x^2+2x+5)-2x(x^2+2x+5)+(p-1)(x^2+2x+5)-2(p-1).x-5(p-1)+10x+q.

=(x^2+2x+5) (x^2–2x+p) +2(6-p).x+5(1-p)+q.

=Divisor × Q +R.

Remainder = 0

2(6-p).x+(5–5p+q)= 0.

2(6-p)x+(5–5p+q)= 0.x + 0.

Equating the coeff. of x and constant term.

2(6-p) = 0 => p = 6 and

5–5p+q = 0.

5–5×6+q = 0.

q = 30–5 = 25

p = 6 and q = 25 , Answer.