In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x^4+10x^3+5x^2+x+1;f(− 2/3)

Verified step by step guidance

1. Step 1: Write down the coefficients of the polynomial function. For the function f(x) = 6x^4 + 10x^3 + 5x^2 + x + 1, the coefficients are 6, 10, 5, 1, and 1.
2. Step 2: Write down the value of x that you are trying to evaluate. In this case, x = -2/3.
3. Step 3: Set up the synthetic division table. Write the value of x on the left, then draw a vertical line. Write the coefficients of the polynomial on the right of the line, then draw a horizontal line underneath.
4. Step 4: Perform synthetic division. Bring down the first coefficient (6 in this case) to the bottom row. Multiply this number by x (-2/3), and write the result under the second coefficient. Add these two numbers and write the result in the bottom row. Repeat this process for all coefficients.
5. Step 5: The last number in the bottom row is the remainder. According to the Remainder Theorem, this is the value of the function at x = -2/3. So, f(-2/3) is equal to the remainder.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a linear binomial of the form (x - c). It involves using the coefficients of the polynomial and performing a series of multiplications and additions to find the quotient and remainder efficiently. This technique is particularly useful for evaluating polynomials at specific values, as it reduces the computational complexity compared to traditional long division.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor (x - c), the remainder of this division is equal to f(c). This theorem allows us to evaluate the polynomial at a specific point without performing the entire division process. In this case, it helps in finding the value of f(-2/3) directly by applying synthetic division.

Polynomial Evaluation

Polynomial evaluation involves substituting a specific value for the variable in a polynomial expression to determine its output. This process can be done using various methods, including direct substitution, synthetic division, or the Remainder Theorem. Understanding how to evaluate polynomials is essential for solving problems in algebra, as it allows for the analysis of function behavior at specific points.