Questions: Consider the following function. Function (f(x)=6 x^3+53 x^2-11 x-18) Factors ((2 x+1),(3 x-2)) (a) Verify the given factors of (f(x)). ((3 x-2) quad frac23 leftlvert, 6 quad 53 quad -11 quad -18 4 quad 38 quad square right. 657 quad square quad square) (b) Find the remaining factor(s) of (f(x)). (Enter your answers as a comma-separated list.)

Solution Steps

To verify the given factors of the polynomial \( f(x) = 6x^3 + 53x^2 - 11x - 18 \), we can use synthetic division. If a factor divides the polynomial without a remainder, then it is indeed a factor. For part (a), we will perform synthetic division using the factor \( 3x - 2 \) and check if the remainder is zero. For part (b), after verifying the factors, we will divide the polynomial by the product of the given factors to find the remaining factor(s).

To verify if \(3x - 2\) is a factor of the polynomial \(f(x) = 6x^3 + 53x^2 - 11x - 18\), we perform synthetic division using the root \(\frac{2}{3}\). The remainder of the division is 0, confirming that \(3x - 2\) is indeed a factor.

Next, we verify if \(2x + 1\) is a factor by performing synthetic division using the root \(-\frac{1}{2}\). The remainder is also 0, confirming that \(2x + 1\) is a factor of the polynomial.

After confirming the factors \(3x - 2\) and \(2x + 1\), we find the remaining factor by dividing the polynomial by the product of these factors. The product of the factors is \(6x^2 + x - 2\). Performing synthetic division, the remaining factor is found to be \(6x + 54\).

Final Answer