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)
f(x)=2x^3+px^2+qx+18
f(1) = 2 + p + q + 18 = 10
p + q = -10
f(-1) = -2 + p - q + 18 = 12
p - q = -4
add the two p&q equations
2p = -14
p = -7, then in your head q = -3
for the roots
f(x)=2x^3 - 7x^2 - 3x + 18 , we already know x≠ ± 1
f(2) = 16 - 28 - 6 + 18 = 0 , YEAHH
so x= 2, and x-2 is a factor
by division:
2x^3 - 7x^2 - 3x + 18 = (x-2)(2x^2 - 3x - 9)
= (x-2)(x - 3)(2x + 3)
so the zeros are 2, 3, -3/2
To find the values of p and q, we can use the Remainder theorem which states that if a polynomial f(x) is divided by x - c, the remainder is equal to f(c).
Given that the remainder when f(x) is divided by (x-1) is 10, we can substitute x = 1 into f(x) and set it equal to 10:
f(1) = 2(1)^3 + p(1)^2 + q(1) + 18 = 10
Simplifying this equation:
2 + p + q + 18 = 10
p + q = -10
Similarly, given that the remainder when f(x) is divided by (x+1) is 12, we can substitute x = -1 into f(x) and set it equal to 12:
f(-1) = 2(-1)^3 + p(-1)^2 + q(-1) + 18 = 12
Simplifying this equation:
-2 + p - q + 18 = 12
p - q = -8
Now we have two equations:
p + q = -10
p - q = -8
Solving these simultaneous equations, we can add the two equations together:
(p + q) + (p - q) = -10 + (-8)
2p = -18
p = -9
Substitute the value of p into one of the equations:
-9 + q = -10
q = -10 + 9
q = -1
Therefore, the values of p and q are p = -9 and q = -1.
To find the zeros of f(x), we need to solve the equation f(x) = 0. We can factorize the polynomial first.
f(x) = 2x^3 + px^2 + qx + 18
f(x) = (x - 1)(x + 1)(2x - 18)
Setting each factor equal to zero:
x - 1 = 0 ---> x = 1
x + 1 = 0 ---> x = -1
2x - 18 = 0 ---> 2x = 18 ---> x = 9
Therefore, the zeros of f(x) are x = 1, x = -1, and x = 9.
To find the values of p and q, we can use the Remainder Theorem. According to the Remainder Theorem, when a polynomial f(x) is divided by (x-a), the remainder is equal to f(a).
In this case, when f(x) is divided by (x-1), the remainder is 10. So, we can substitute x=1 into f(x) and set it equal to 10:
f(1) = 2(1)^3 + p(1)^2 + q(1) + 18 = 10
Simplifying the equation, we have:
2 + p + q + 18 = 10
p + q = -10 ----(1)
Similarly, when f(x) is divided by (x+1), the remainder is 12. So, we can substitute x=-1 into f(x) and set it equal to 12:
f(-1) = 2(-1)^3 + p(-1)^2 + q(-1) + 18 = 12
Simplifying the equation, we have:
-2 + p - q + 18 = 12
p - q = -8 ----(2)
Now, we have a system of equations with p + q = -10 (equation 1) and p - q = -8 (equation 2). We can solve this system of equations to find the values of p and q.
Adding equation 1 and equation 2, we get:
2p = -18
Dividing both sides by 2, we find:
p = -9
Substituting the value of p into equation 1, we can find q:
-9 + q = -10
q = -1
Therefore, the values of p and q are: p = -9 and q = -1.
To find the zeros of f(x), we need to solve the equation f(x) = 0.
f(x) = 2x^3 + px^2 + qx + 18
Substituting the values of p and q we found earlier, the polynomial becomes:
f(x) = 2x^3 - 9x^2 - x + 18
Now, we can set f(x) = 0 and solve for x:
2x^3 - 9x^2 - x + 18 = 0
To solve this equation, you can use various methods such as factoring, synthetic division, or using a graphing calculator. The solutions or zeros of f(x) are the values of x that make the equation equal to zero.