The remainder when the polynomial f(x)=2x3+px2+qx+18 is divided by (x-1) is 10, when it is divided by (x+1) the remainder is R find
(1) The values of p and q
(2) The zeros of f(x)
p=-7 q=-3
The answe
-4and-7
To find the values of p and q, we can make use of the Remainder Theorem. According to this theorem, when a polynomial f(x) is divided by x - c, the remainder is equal to f(c).
Using this information, we can substitute x = 1 into the polynomial f(x) and set it equal to 10, the remainder when divided by (x - 1). So we have:
f(1) = 10
2(1)^3 + p(1)^2 + q(1) + 18 = 10
2 + p + q + 18 = 10
p + q = -10
Next, to find the value of R (the remainder when divided by (x + 1)), we can substitute x = -1 into f(x) and set it equal to R. So we have:
f(-1) = R
2(-1)^3 + p(-1)^2 + q(-1) + 18 = R
-2 + p - q + 18 = R
p - q + 16 = R
So, we have the following equations:
p + q = -10 (equation 1)
p - q + 16 = R (equation 2)
Now we can solve these equations to find the values of p, q, and R.
To find the zeros of f(x), we set f(x) equal to zero and solve for x. So we have:
2x^3 + px^2 + qx + 18 = 0
Unfortunately, there isn't enough information provided to directly solve for the zeros of f(x) and find the exact values. Additional information or context would be needed to proceed with solving for the zeros.
However, once we obtain the values of p and q from the first step, we can substitute them back into the equation and use mathematical techniques such as factoring, long division, or synthetic division to solve for the zeros of f(x).