Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=3x³−7x²−2x+5;f(−3)

Verified step by step guidance

1. Identify the divisor for synthetic division based on the value at which the function is evaluated. Since we want to find \( f(-3) \), the divisor is \( x - (-3) = x + 3 \).
2. Set up synthetic division by writing the coefficients of the polynomial \( f(x) = 3x^{3} - 7x^{2} - 2x + 5 \) in order: \( 3, -7, -2, 5 \).
3. Write \(-3\) (the zero of the divisor \( x + 3 \)) to the left and perform synthetic division: bring down the first coefficient, multiply by \(-3\), add to the next coefficient, and repeat this process for all coefficients.
4. The final number obtained after completing synthetic division is the remainder, which by the Remainder Theorem equals \( f(-3) \).
5. Interpret the remainder as the value of the function at \( x = -3 \), which completes the evaluation.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This theorem allows us to find the value of the polynomial at x = c without fully performing the division.

Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value means substituting that value for the variable and calculating the result. Using synthetic division and the Remainder Theorem provides an efficient way to find f(c) without direct substitution.