Asked by Benedict
The quadratic function ๐(๐ฅ) which is exactly divisible by 2๐ฅ + 1 and has remainder -6 and -5
when divided by ๐ฅ โ 1 and ๐ฅ โ 2 respectively. Determine g(x)=(๐๐ฅ + ๐)2๐(๐ฅ) where p and q
are constants and equal. Given that on division by (๐ฅ โ 2)2,๐กโ๐ ๐๐๐๐๐๐๐๐๐ โ 39 โ 3๐ฅ
when divided by ๐ฅ โ 1 and ๐ฅ โ 2 respectively. Determine g(x)=(๐๐ฅ + ๐)2๐(๐ฅ) where p and q
are constants and equal. Given that on division by (๐ฅ โ 2)2,๐กโ๐ ๐๐๐๐๐๐๐๐๐ โ 39 โ 3๐ฅ
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.
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.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.