If (x-3) is a factor of 3x^3+bx^2-17x-12.
a. Find the value of b.
b.Hence , find the roots fo the equation 3x^3+bx^2-17x-12=0.
if x=3, then f(x)=0, so put in x=3 into f(x)=0 and solve for b.
Now that you know b, you can divide out the factor (x-2) and are left with a quadratic 3x^2+ ... which if =0, then you can factor or use the quadratic equation to find those two roots.
To find the value of b, we can use the fact that (x - 3) is a factor of 3x^3 + bx^2 - 17x - 12. Since (x - 3) is a factor, it means that if we substitute x = 3 into the expression 3x^3 + bx^2 - 17x - 12, the result will be zero.
a. Find the value of b:
Substituting x = 3 into the equation 3x^3 + bx^2 - 17x - 12 = 0:
3(3)^3 + b(3)^2 - 17(3) - 12 = 0.
27 + 9b - 51 - 12 = 0.
9b - 36 = 0.
9b = 36.
b = 4.
Therefore, the value of b is 4.
b. Finding the roots of the equation:
Now that we know b = 4, we can proceed to find the roots of the equation 3x^3 + 4x^2 - 17x - 12 = 0. One way to find these roots is by using synthetic division, which can help us factorize the cubic equation.
Using synthetic division, we divide the cubic equation by (x - 3):
3 | 3 4 -17 -12
| - 9 39 66
---------------------
| 3 -5 22 54
The quotient after synthetic division is 3x^2 - 5x + 22 with a remainder of 54.
So, the factored form of the equation 3x^3 + 4x^2 - 17x - 12 = 0 is:
(x - 3)(3x^2 - 5x + 22) = 0.
Now, we need to find the roots of the quadratic equation 3x^2 - 5x + 22 = 0. Let's solve this quadratic equation by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a.
For our equation 3x^2 - 5x + 22 = 0, the values of a, b, and c are:
a = 3, b = -5, c = 22.
Now, substituting these values into the quadratic formula:
x = (-(-5) ± √((-5)^2 - 4(3)(22))) / (2(3))
= (5 ± √(25 - 264)) / 6
= (5 ± √(-239)) / 6.
Since √(-239) is not a real number, it means that the quadratic equation has no real solutions. Therefore, the equation 3x^3 + 4x^2 - 17x - 12 = 0 has only one real root, which is x = 3.