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For x2+2x+5 to be a factor of x4+ px2+q, what must the respective values of p and q be?
- a)−2 and 5
- b)5 and 25
- c)10 and 20
- d)6 and 25
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Forx2+2x+5 to be a factor ofx4+px2+q, what must the respective values ...
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.
Most Upvoted Answer
Forx2+2x+5 to be a factor ofx4+px2+q, what must the respective values ...
To determine the values of p and q, we need to use the factor theorem. According to the factor theorem, if x - a is a factor of a polynomial, then the polynomial evaluated at a will equal zero.
In this case, since 2x + 2 is a factor, we have:
(2x + 2) * (x^3 + px^2 + qx - r) = 0
To find the values of p and q, we can substitute x = -1 into the equation:
(2(-1) + 2) * ((-1)^3 + p(-1)^2 + q(-1) - r) = 0
Simplifying:
(-2 + 2) * (-1 + p + q - r) = 0
0 * (-1 + p + q - r) = 0
This equation tells us that the expression inside the parentheses must equal zero for the polynomial to be divisible by 2x + 2.
Therefore, -1 + p + q - r = 0
Since we don't have enough information to determine the value of r, we can't find the exact values of p and q.
In this case, since 2x + 2 is a factor, we have:
(2x + 2) * (x^3 + px^2 + qx - r) = 0
To find the values of p and q, we can substitute x = -1 into the equation:
(2(-1) + 2) * ((-1)^3 + p(-1)^2 + q(-1) - r) = 0
Simplifying:
(-2 + 2) * (-1 + p + q - r) = 0
0 * (-1 + p + q - r) = 0
This equation tells us that the expression inside the parentheses must equal zero for the polynomial to be divisible by 2x + 2.
Therefore, -1 + p + q - r = 0
Since we don't have enough information to determine the value of r, we can't find the exact values of p and q.
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