The function f(x) = ax^3-x^2+bx-24 has three factors. Two of these factors are x-2 and x+4. Determine the values of a and b, and then determine the other factor
sorry buddy im stuck too.
To find the values of a and b, we can use the fact that the given function has three factors, two of which are (x-2) and (x+4).
Step 1: Multiply the two known factors: (x-2) * (x+4) = x^2 + 2x - 8.
Step 2: Divide the original function f(x) by the product of the known factors:
f(x) / [(x-2) * (x+4)] = (ax^3 - x^2 + bx - 24) / (x^2 + 2x - 8)
Step 3: Set this quotient equal to the unknown factor:
(ax^3 - x^2 + bx - 24) / (x^2 + 2x - 8) = x + c
Step 4: Multiply both sides by the denominator (x^2 + 2x - 8) to get rid of the fraction:
(ax^3 - x^2 + bx - 24) = (x + c)(x^2 + 2x - 8)
Step 5: Expand the right side of the equation:
ax^3 - x^2 + bx - 24 = x^3 + cx^2 + 2cx - 8x + cx + 2c - 8
Step 6: Combine like terms on both sides:
ax^3 - x^2 + bx - 24 = x^3 + cx^2 + (2c - 8)x + (cx + 2c - 8)
Step 7: Compare the coefficients of each term on both sides:
For the x^3 term: a = 1.
For the x^2 term: -1 = c.
For the x term: b = 2c - 8.
For the constant term: -24 = cx + 2c - 8.
Step 8: Solve these equations to find the values of a, b, and c:
From equation 1, a = 1.
From equation 2, c = -1.
From equation 3, b = 2c - 8 = 2*(-1) - 8 = -2 - 8 = -10.
So, a = 1, b = -10, and c = -1.
To determine the other factor, substitute the values of a, b, and c back into the equation from Step 3:
f(x) / [(x-2) * (x+4)] = (1x^3 - x^2 - 10x - 24) / (x^2 + 2x - 8) = x - 6.
Therefore, the other factor is (x - 6).