When dividedby x+2, the expression ax^3+bx^2-cx+4 leaves remainder -22 when divided by x-1, the same expression leaves remainder 2. Given that when the expression is divided by x-2, the remainder is 6. Find a, b, c.
When the expression is divided by x+2, we have:
ax^3 + bx^2 - cx + 4 = q(x)(x+2) - 22
Where q(x) is the quotient. When the expression is divided by x-1, we have:
ax^3 + bx^2 - cx + 4 = p(x)(x-1) + 2
Where p(x) is the quotient. And when the expression is divided by x-2, we have:
ax^3 + bx^2 - cx + 4 = r(x)(x-2) + 6
Where r(x) is the quotient.
We can see that the remainder when divided by x+2 is -22, so q(-2) = -22. We can also see that the remainder when divided by x-1 is 2, so p(1) = 2. And finally, the remainder when divided by x-2 is 6, so r(2) = 6.
To find a, b, and c, we need to solve these equations.
q(-2) = -22
p(1) = 2
r(2) = 6
Expanding these equations, we get:
-8a + 4b -2c + 4 = -22
a + b - c + 4 = 2
4a + 2b - 4c + 4 = 6
Simplifying these equations, we get:
-8a + 4b -2c = -26
a + b - c = -2
4a + 2b - 4c = 2
Now, we can solve these equations using any method of solving a system of equations, such as substitution or elimination. For simplicity, we will use elimination method.
Multiplying the second equation by 4, we get:
4a + 4b - 4c = -8
4a + 2b - 4c = 2
Subtracting the second equation from the first equation, we get:
2b = -10
b = -5
Substituting b = -5 into the second equation, we get:
a + (-5) - c = -2
a - c = 3
Substituting b = -5 into the third equation, we get:
4a + 2(-5) - 4c = 2
4a - 10 - 4c = 2
4a - 4c = 12
Subtracting the second equation from the third equation, we get:
4c = -9
c = -9/4
Substituting c = -9/4 into the second equation, we get:
a - (-9/4) = 3
a + 9/4 = 3
a = 3 - 9/4
a = 3/4
So, a = 3/4, b = -5, and c = -9/4.
To find the values of a, b, and c, we need to use the remainder theorem. The remainder theorem states that if a polynomial f(x) is divided by x - r, the remainder is equal to f(r).
Let's apply the remainder theorem to the given expression:
1. Dividing ax^3 + bx^2 - cx + 4 by x + 2:
The remainder is given as -22.
Using the remainder theorem, we can set up the equation:
(a(1)^3 + b(1)^2 - c(1) + 4) = -22
Simplifying, we have:
a + b - c + 4 = -22 (Equation 1)
2. Dividing ax^3 + bx^2 - cx + 4 by x - 1:
The remainder is given as 2.
Using the remainder theorem, we can set up the equation:
(a(-1)^3 + b(-1)^2 - c(-1) + 4) = 2
Simplifying, we have:
-a + b + c + 4 = 2 (Equation 2)
3. Dividing ax^3 + bx^2 - cx + 4 by x - 2:
The remainder is given as 6.
Using the remainder theorem, we can set up the equation:
(a(2)^3 + b(2)^2 - c(2) + 4) = 6
Simplifying, we have:
8a + 4b - 2c + 4 = 6 (Equation 3)
Now, we have three equations (Equation 1, 2, and 3) with three variables (a, b, and c). We can solve this system of equations to find the values of a, b, and c.
Let's solve the equations:
From Equation 1, we have:
a + b - c + 4 = -22 --> a + b - c = -26 (Equation 4)
From Equation 2, we have:
-a + b + c + 4 = 2 --> -a + b + c = -2 (Equation 5)
From Equation 3, we have:
8a + 4b - 2c + 4 = 6 --> 8a + 4b - 2c = 2 (Equation 6)
Now we have a system of three equations (Equation 4, 5, and 6) that we can solve simultaneously.
Solving the system of equations:
We can solve this system of equations using any method such as substitution, elimination, or matrix methods.
Let's solve using the elimination method:
Multiplying Equation 4 by 4, we have:
4(a + b - c) = 4(-26) --> 4a + 4b - 4c = -104 (Equation 7)
Adding Equation 5 and Equation 7:
(-a + b + c) + (4a + 4b - 4c) = (-2) + (-104)
3a + 5b = -106 (Equation 8)
Multiplying Equation 5 by 8, we have:
8(-a + b + c) = 8(-2) --> -8a + 8b + 8c = -16 (Equation 9)
Adding Equation 6 and Equation 9:
(8a + 4b - 2c) + (-8a + 8b + 8c) = 2 + (-16)
12b + 6c = -14 (Equation 10)
We now have a new system of equations (Equation 8 and 10) with two variables (b and c).
Let's solve this system of equations:
Multiplying Equation 8 by 2, we have:
2(3a + 5b) = 2(-106) --> 6a + 10b = -212 (Equation 11)
Multiplying Equation 10 by 5, we have:
5(12b + 6c) = 5(-14) --> 60b + 30c = -70 (Equation 12)
Adding Equation 11 and Equation 12:
(6a + 10b) + (60b + 30c) = (-212) + (-70)
6a + 70b + 30c = -282
Simplifying, we have:
6a + 70b + 30c = -282 (Equation 13)
We now have Equation 13 with variables a, b, and c. However, we still need one more equation to solve this system.
At this point, we need additional information or an equation to solve for a, b, and c. Without any further information or equations, we cannot determine the values of a, b, and c.
To find the values of a, b, and c, 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).
We are given that when the expression ax^3 + bx^2 - cx + 4 is divided by x + 2, the remainder is -22. Using the remainder theorem, we have:
f(-2) = a(-2)^3 + b(-2)^2 - c(-2) + 4 = -22
Simplifying the equation:
-8a + 4b + 2c + 4 = -22
-8a + 4b + 2c = -26 .........(1)
We are also given that when the expression is divided by x - 1, the remainder is 2. Using the remainder theorem again, we have:
f(1) = a(1)^3 + b(1)^2 - c(1) + 4 = 2
Simplifying the equation:
a + b - c + 4 = 2
a + b - c = -2 ............(2)
Finally, we are given that when the expression is divided by x - 2, the remainder is 6. Using the remainder theorem one more time, we have:
f(2) = a(2)^3 + b(2)^2 - c(2) + 4 = 6
Simplifying the equation:
8a + 4b - 2c + 4 = 6
8a + 4b - 2c = 2 ............(3)
Now we have a system of three linear equations with three unknowns (a, b, and c). We can solve this system to find the values of a, b, and c.
By solving equations (1), (2), and (3) simultaneously, we will find the values of a, b, and c.