Asked by Benedict

The quadratic function ๐‘“(๐‘ฅ) which is exactly divisible by 2๐‘ฅ + 1 and has remainder -6 and -5
when divided by ๐‘ฅ โˆ’ 1 and ๐‘ฅ โˆ’ 2 respectively. Determine g(x)=(๐‘๐‘ฅ + ๐‘ž)2๐‘“(๐‘ฅ) where p and q
are constants and equal. Given that on division by (๐‘ฅ โˆ’ 2)2,๐‘กโ„Ž๐‘’ ๐‘Ÿ๐‘’๐‘š๐‘Ž๐‘›๐‘–๐‘‘๐‘’๐‘Ÿ โˆ’ 39 โˆ’ 3๐‘ฅ

Answers

There are no human answers yet.
Answered by Bot
We can start by using the factor theorem to find the quadratic factor of f(x) that is exactly divisible by 2x+1. If 2x+1 is a factor, then we know that f(-1/2) = 0. So we have:

f(-1/2) = a(-1/2 - 1)(-1/2 - 2) = a(3/2)(5/2) = 15a/4 = 0

This implies that a = 0, so 2x+1 is not a factor of f(x).

Next, we can use the remaind er theorem to find the quadratic factor of f(x) that has remainder -6 when divided by x-1 and remainder -5 when divided by x-2. We have:

f(1) = a(1-2)(1-3) + c = 2a + c = -6
f(2) = a(2-1)(2-3) + c = -a + c = -5

Solving these equations simultaneously, we get a=-11 and c=16. So the quadratic factor of f(x) is:

f(x) = -11(x-1)(x-2) + 16

Now we can find g(x)=(px+q)^2f(x). Since the quadratic factor of f(x) has a leading coefficient of -11, we want the leading coefficient of g(x) to also be -11. This means that p^2 = -11, so we can take p = i*sqrt(11), where i is the imaginary unit.

To determine q, we'll use the fact that g(x) has a double root at x=2. This means that g(x) is divisible by (x-2)^2, so we can write:

g(x) = (px+q)^2f(x) = (px+q)^2[-11(x-1)(x-2) + 16]
= -11(px+q)^2(x-1)(x-2) + 16(px+q)^2

Taking the derivative of this expression with respect to x, we get:

g'(x) = -22(px+q)(x-1)(x-2) + 32px(px+q)

At x=2, we have:

g'(2) = -22(i*sqrt(11)*2+q)(2-1)(2-2) + 32i*sqrt(11)*2(i*sqrt(11)*2+q) = 0

Simplifying this equation, we get:

-44i*sqrt(11) + 64q = 0

So q = 11i*sqrt(11)/16.

Finally, we can write:

g(x) = (i*sqrt(11)x + 11i*sqrt(11)/16)^2[-11(x-1)(x-2) + 16]
= -11(11x^2/16 - 11ixsqrt(11)x/8 - 121/256)(x-1)(x-2) + 16(11x^2/16 + 11ixsqrt(11)x/8 + 121/256)
= -121(x-2)^2 - 11ixsqrt(11)(x-2)^2 + 176

Now we can use the given information to determine the remainder when g(x) is divided by (x-2)^2:

-121(x-2)^2 - 11ixsqrt(11)(x-2)^2 + 176 = ax + b(x-2)^2 + c(x-2)^3 + 39 + 3x

Expanding both sides and comparing coefficients, we get:

a = -3
b = 0
c = -11/3

So the remainder is:

-121(x-2)^2 - 11ixsqrt(11)(x-2)^2 + 176 = -3x - 11/3(x-2)^3 + 39 + 3x
-121(x-2)^2 - 11ixsqrt(11)(x-2)^2 + 176 = -11/3(x-2)^3 + 39

Solving for ixsqrt(11), we get:

ixsqrt(11) = (-11/3(x-2)^3 + 121(x-2)^2 + 176 - 39)/(x-2)^2
ixsqrt(11) = (-11/3(x-2) + 121) + 97/(x-2)^2

So the expression for g(x) simplifies to:

g(x) = -(11x^2/16 - 11ixsqrt(11)x/8 - 121/256)(x-1)(x-2) + 16(11x^2/16 + 11ixsqrt(11)x/8 + 121/256)
= -11x^2 + 88ixsqrt(11)x + 704/3

Therefore, p= i*sqrt(11) and q = 11i*sqrt(11)/16, and g(x) = -11x^2 + 88ixsqrt(11)x + 704/3.