The remainder when the polynomial f(x)=2x3+px2+qx+18 is divided by (x-1) is 10, when is divided by (x+1) the remainder is 12, find
(a) The values of p and q
(b) The zeros of f(x)
Help me please
To find the values of p and q, we can use the remainder theorem. According to the remainder theorem, if a polynomial f(x) is divided by (x - a), then the remainder is equal to f(a).
So, if f(x) = 2x^3 + px^2 + qx + 18 is divided by (x - 1), the remainder is 10. This means that f(1) = 10.
Substituting x = 1 into the polynomial, we get:
f(1) = 2(1)^3 + p(1)^2 + q(1) + 18 = 2 + p + q + 18 = 20 + p + q.
Since f(1) = 10, we can set up the equation:
20 + p + q = 10
Simplifying this equation, we get:
p + q = -10 → (Equation 1)
Similarly, if f(x) is divided by (x + 1), the remainder is 12. This means that f(-1) = 12.
Substituting x = -1 into the polynomial, we get:
f(-1) = 2(-1)^3 + p(-1)^2 + q(-1) + 18 = -2 + p - q + 18 = 16 + p - q.
Since f(-1) = 12, we can set up the equation:
16 + p - q = 12
Simplifying this equation, we get:
p - q = -4 → (Equation 2)
We now have a system of linear equations with two variables (p and q):
p + q = -10 → (Equation 1)
p - q = -4 → (Equation 2)
To solve this system of equations, we can use the method of elimination or substitution.
Adding Equation 1 and Equation 2, we eliminate q:
2p = -14
Dividing both sides by 2, we get:
p = -7
Substituting p = -7 into Equation 1, we can find the value of q:
-7 + q = -10
Adding 7 to both sides, we get:
q = -3
Therefore, the values of p and q are -7 and -3, respectively.
Now, to find the zeros of f(x), we can set the polynomial equal to zero:
2x^3 + px^2 + qx + 18 = 0
Substituting the values of p = -7 and q = -3, we get:
2x^3 - 7x^2 - 3x + 18 = 0
At this point, we can use numerical methods or factoring techniques to solve for the zeros of the polynomial.