Given the f (x) =2x^3 + ax^2 + bx - 9. (a)Find the value of a and b if f (x) has a factor x -3 but leaves a reminder of 8 when divided by x + 1. (b) Factorize f (x) completely and hence solve for x.
To find the value of a and b in the equation f(x) = 2x^3 + ax^2 + bx - 9, we can use the information given about the factors and remainders.
(a) Find the value of a and b if f(x) has a factor x - 3 but leaves a remainder of 8 when divided by x + 1.
1. If f(x) has a factor of x - 3, it means that f(3) should be equal to 0 because when x = 3, the factor (x - 3) becomes zero. So, let's substitute x = 3 into the equation f(x) and set it equal to 0:
f(3) = 2(3)^3 + a(3)^2 + b(3) - 9 = 0
Simplifying the equation:
54 + 9a + 3b - 9 = 0
45 + 9a + 3b = 0
9a + 3b = -45 (equation 1)
2. If f(x) leaves a remainder of 8 when divided by x + 1, it means that f(-1) should be equal to 8 because when x = -1, the remainder is 8. So, let's substitute x = -1 into the equation f(x) and set it equal to 8:
f(-1) = 2(-1)^3 + a(-1)^2 + b(-1) - 9 = 8
Simplifying the equation:
-2 + a - b - 9 = 8
a - b = 19 (equation 2)
Now we have two equations (equation 1 and equation 2) with two variables (a and b). We can solve these equations simultaneously to find the values of a and b.
Solving equations 1 and 2 simultaneously:
Multiply equation 2 by 3 to make the coefficients of b the same and eliminate b:
9a + 3b = -45 (equation 1)
3a - 3b = 57 (equation 2)
Adding equation 1 and equation 2 together:
9a + 3b + 3a - 3b = -45 + 57
12a = 12
a = 1
Substitute the value of a = 1 into equation 2 to solve for b:
1 - b = 19
b = -18
Therefore, the values of a and b are a = 1 and b = -18.
(b) Factorize f(x) completely and hence solve for x.
Now that we have the values of a and b, we can factorize f(x) using the given quadratic expression.
f(x) = 2x^3 + ax^2 + bx - 9
Substituting the values of a = 1 and b = -18 into the equation:
f(x) = 2x^2 + x^2 - 18x - 9
Factoring out the common factor (x - 3):
f(x) = (x - 3)(2x^2 + 3x + 3)
The quadratic term (2x^2 + 3x + 3) cannot be factored further, so the complete factorization of f(x) is:
f(x) = (x - 3)(2x^2 + 3x + 3)
To solve for x, we need to set each factor equal to zero and solve for x:
Setting x - 3 = 0:
x - 3 = 0
x = 3
Setting 2x^2 + 3x + 3 = 0:
Unfortunately, the quadratic term does not have any real solutions. Therefore, the solution to the equation f(x) = 0 is x = 3.