i got p to be 11
but i do not know how to solve the equation
6x^3 - 11x^2 - 14x + 24=0
Now, you know one root is x=2, so that is a factor. Divide x-2 into the polynomial.
That will give you a binomial, which will give the other two factors.
To solve the equation 6x^3 - 11x^2 - 14x + 24 = 0, you have mentioned that one root is x = 2. This means that (x - 2) is a factor of the polynomial.
To find the remaining factors, you can perform polynomial division. Here's how you can do it:
1. Write the polynomial in descending order of exponents:
6x^3 - 11x^2 - 14x + 24 = 0
2. Set up the polynomial division by placing the divisor (x - 2) outside the long division symbol and the dividend (6x^3 - 11x^2 - 14x + 24) inside the symbol.
_____________________
x - 2 | 6x^3 - 11x^2 - 14x + 24
3. Divide the first term of the dividend (6x^3) by the first term of the divisor (x). This gives you 6x^2. Place this result above the horizontal line.
6x^2
_____________________
x - 2 | 6x^3 - 11x^2 - 14x + 24
4. Multiply the divisor (x - 2) by the result (6x^2), which gives you 6x^3 - 12x^2. Write this term below the dividend and subtract it from the corresponding terms.
6x^2
_____________________
x - 2 | 6x^3 - 11x^2 - 14x + 24
- (6x^3 - 12x^2)
= x^2 - 14x + 24
5. Repeat steps 3 and 4 with the new polynomial (x^2 - 14x + 24) as the dividend.
6x^2 + x - 2
_____________________
x - 2 | 6x^3 - 11x^2 - 14x + 24
- (6x^3 - 12x^2)
_____________________
- x^2 - 14x + 24
6. Divide the first term of the new dividend (-x^2) by the first term of the divisor (x). This gives you -x. Place this result above the horizontal line.
6x^2 + x - 2
_____________________
x - 2 | 6x^3 - 11x^2 - 14x + 24
- (6x^3 - 12x^2)
_____________________
- x^2 - 14x + 24
- (-x^2 + 2x)
= -12x + 24
7. Repeat steps 3 and 4 with the new polynomial (-x^2 - 14x + 24) as the dividend.
6x^2 + x - 2 - 12
_____________________
x - 2 | 6x^3 - 11x^2 - 14x + 24
- (6x^3 - 12x^2)
_____________________
- x^2 - 14x + 24
- (-x^2 + 2x)
_____________________
- 12x + 24
8. Divide the first term of the new dividend (-12x) by the first term of the divisor (x). This gives you -12. Place this result above the horizontal line.
6x^2 + x - 2 - 12
_____________________
x - 2 | 6x^3 - 11x^2 - 14x + 24
- (6x^3 - 12x^2)
_____________________
- x^2 - 14x + 24
- (-x^2 + 2x)
_____________________
- 12x + 24
- (-12x + 24)
= 0
9. Now, the result at the top of the horizontal line represents the quotient of the polynomial division. In this case, it is 6x^2 + x - 2.
So, factoring the polynomial 6x^3 - 11x^2 - 14x + 24 = 0, we have:
(x - 2)(6x^2 + x - 2) = 0
To find the remaining factors, you can solve the quadratic equation 6x^2 + x - 2 = 0 using any appropriate method, such as factoring, completing the square, or using the quadratic formula.