the polynomial f(x)=a(x-1)^2+b(x+2)^2 is divided by x+1 and x-2 the remainders are 3 and-15 respectively find the value of a and b
f(x)=a(x-1)^2+b(x+2)^2
given:
f(-1) = 3
a(4) + b(1) = 3 -----> 4a + b = 3
f(2) = -15
a + 16b = -15 or a = -15-16b
substitute:
4(-15-16b) + b = 3
-60 - 64b + b = 3
-63b = 63
b = -1 , then a = -15 + 16 = 1
To find the values of a and b, we can use the remainder theorem. According to the remainder theorem, when a polynomial f(x) is divided by x - c, the remainder is equal to f(c).
Given that:
- The polynomial f(x) = a(x-1)^2 + b(x+2)^2
- The remainder when f(x) is divided by x + 1 is 3
- The remainder when f(x) is divided by x - 2 is -15
Let's substitute the values of x for each case:
1) When f(x) is divided by x + 1:
f(-1) = (a(-1-1)^2 + b(-1+2)^2
3 = 4a + b
2) When f(x) is divided by x - 2:
f(2) = a(2-1)^2 + b(2+2)^2
-15 = a + 16b
Now we have a system of equations:
4a + b = 3 ---- (Equation 1)
a + 16b = -15 ---- (Equation 2)
We can solve this system to find the values of a and b. Subtracting Equation 1 from Equation 2:
(a + 16b) - (4a + b) = -15 - 3
a + 16b - 4a - b = -18
-3a + 15b = -18
Dividing both sides of the equation by -3, we get:
a - 5b = 6 ---- (Equation 3)
Now we can solve the system of equations formed by Equation 1 and Equation 3.
Multiply Equation 1 by 5:
20a + 5b = 15 ---- (Equation 4)
Now subtract Equation 3 from Equation 4:
(20a + 5b) - (a - 5b) = 15 - 6
20a + 5b - a + 5b = 9
19a + 10b = 9
Dividing both sides by 19, we get:
a + (10/19)b = 9/19 ---- (Equation 5)
Now we have two equations:
a - 5b = 6 ---- (Equation 3)
a + (10/19)b = 9/19 ---- (Equation 5)
Multiply Equation 3 by 10 and Equation 5 by 19 to eliminate decimals:
10a - 50b = 60 ---- (Equation 6)
19a + 10b = 9 ---- (Equation 7)
Add Equation 6 and Equation 7:
(10a - 50b) + (19a + 10b) = 60 + 9
29a - 40b = 69
Divide both sides by 29:
a - (40/29)b = 69/29
We have simplified the equations, but the values of a and b are not yet determined. To find the exact values, we need another equation or piece of information.
To find the values of "a" and "b," we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial f(x) is divided by x - c, then the remainder is given by f(c).
Given that the polynomial f(x) = a(x - 1)^2 + b(x + 2)^2 is divided by x + 1 and x - 2, we have the following information:
Remainder when f(x) is divided by x + 1 = 3
Remainder when f(x) is divided by x - 2 = -15
Using these values, we can set up two equations:
1. f(-1) = 3 ----(equation 1)
2. f(2) = -15 ----(equation 2)
Now, let's calculate f(-1) and f(2) using the given polynomial:
1. For f(-1):
f(-1) = a(-1 - 1)^2 + b(-1 + 2)^2
= a(-2)^2 + b(1)^2
= 4a + b
2. For f(2):
f(2) = a(2 - 1)^2 + b(2 + 2)^2
= a(1)^2 + b(4)^2
= a + 16b
Substituting these values in equations 1 and 2:
1. 4a + b = 3 ----(equation 3)
2. a + 16b = -15 ----(equation 4)
Now, we can solve these two equations to find the values of "a" and "b".
To solve these simultaneous equations, we can use any method. For simplicity, let's solve them using the substitution method:
1. Rearrange equation 3 to solve for "b":
b = 3 - 4a
2. Substitute the value of "b" in equation 4:
a + 16(3 - 4a) = -15
a + 48 - 64a = -15
-63a = -63
a = 1
Substituting the value of "a" into equation 3:
4(1) + b = 3
4 + b = 3
b = -1
Therefore, the values of "a" and "b" are a = 1 and b = -1, respectively.