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.

Answer 1

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.