When the polynomial p(x) is divided by (x–2), the remainder is 3 and when p(x) is divided by (x+1) the remainder is 9. Given that p(x) may be written in the form (x–2)(x+1)q(x) + Ax + B where q(x) is a polynomial and A and B are numbers, find the remainder when p(x) is divided by (x–2)(x+1).

Can someone please actually help?

impatient much? We don't all just sit by our computers all day waiting for postings. We also have lives.

we are given
p(2) = 3
p(-1) = 9

so, using those values (and recognizing that the (x–2)(x+1)q(x) part is zero),

2A+B = 3
-A+B = 9

solve for A and B, and Ax+B is the remainder.

Steve,

My apologies but the only reason I said that was because some random person kept spamming this question with random sayings and it kinda got annoying. I didn't mean to offend you in any way

yeah, that happens some here.

To find the remainder when dividing a polynomial by another polynomial, we can use the Remainder Theorem. The Remainder Theorem states that if we divide a polynomial p(x) by a linear factor (x-x0), where x0 is a number, then the remainder will be equal to p(x0).

In this case, we are given that the remainder when p(x) is divided by (x-2) is 3. This means that p(2) = 3. Similarly, we are given that the remainder when p(x) is divided by (x+1) is 9, so p(-1) = 9.

Now, we can use these two equations to find the values of A and B in the expression (x-2)(x+1)q(x) + Ax + B.

Substituting x = 2 into the expression, we get:
(2-2)(2+1)q(2) + A(2) + B = 3
0(3)q(2) + 2A + B = 3
2A + B = 3 ...(1)

Similarly, substituting x = -1 into the expression, we get:
(-1-2)(-1+1)q(-1) + A(-1) + B = 9
(-3)(0)q(-1) - A + B = 9
B - A = 9 ...(2)

Now, we have a system of two equations with two variables (A and B). Solving equations (1) and (2) simultaneously will give us the values of A and B.

We can add equations (1) and (2) together to eliminate B:
2A + B + B - A = 3 + 9
A = 12

Now, substitute the value of A (12) back into equation (2) to solve for B:
B - 12 = 9
B = 21

Therefore, we have found that A = 12 and B = 21.

Now, substitute the values of A and B back into the expression (x-2)(x+1)q(x) + Ax + B to get the complete polynomial p(x).

p(x) = (x-2)(x+1)q(x) + 12x + 21

Finally, to find the remainder when p(x) is divided by (x-2)(x+1), we can substitute x = -1 and calculate p(-1):

p(-1) = (-1-2)(-1+1)q(-1) + 12(-1) + 21
= 0q(-1) - 12 + 21
= 9

Therefore, the remainder when p(x) is divided by (x-2)(x+1) is 9.