Question

Use the factor theorem to determine if the given binomial x - 3 is a factor of the polynomial f(x) = x⁴ - 3x³ + 6x² + 2x - 60 Show the step-by-step process to evaluate f(3)f(3) and determine whether or not x - 3x−3 is a factor of the polynomial. If it is a factor, explain how you arrived at this conclusion and provide the quotient polynomial.

Answer 1

The factor theorem is used to verify that x - 3 is a factor of the polynomial f(x) by evaluating f(3) to check if it results in zero. Upon calculating, f(3) equals zero, confirming that x - 3 is a factor. Polynomial long division can then be used to find the quotient polynomial.

Step-by-step explanation:

The student has asked to use the factor theorem to determine if x - 3 is a factor of the polynomial f(x) = x⁴ - 3x³ + 6x² + 2x - 60. A step-by-step process involves evaluating f(3) to check if it equals zero.

1. Substitute x = 3 into f(x): f(3) = (3)⁴ - 3(3)³ + 6(3)² + 2(3) - 60.
2. Calculate: f(3) = 81 - 81 + 54 + 6 - 60.
3. When simplified, f(3) = 0, confirming that x - 3 is a factor of the polynomial according to the factor theorem.
4. To find the quotient polynomial, perform polynomial long division of f(x) by x - 3.

Since we found that x - 3 is indeed a factor, it confirms that when f(x) is divided by x - 3, the remainder will be 0, thus yielding a quotient polynomial.