In Exercises 17–32, divide using synthetic division. (x^4−256)/(x−4)

Verified step by step guidance

1. Step 1: Set up the synthetic division. Write down the coefficients of the dividend polynomial, which are 1, 0, 0, 0, and -256, corresponding to x^4, x^3, x^2, x, and the constant term, respectively.
2. Step 2: Identify the zero of the divisor x - 4, which is x = 4. This is the number you will use for synthetic division.
3. Step 3: Begin the synthetic division process. Bring down the leading coefficient (1) to the bottom row.
4. Step 4: Multiply the number you just brought down (1) by the zero of the divisor (4) and write the result under the next coefficient (0). Add this result to the coefficient above it and write the sum below. Repeat this process for each coefficient.
5. Step 5: Continue the process until you have worked through all coefficients. The numbers in the bottom row represent the coefficients of the quotient polynomial, and the last number is the remainder.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing polynomials, particularly useful when dividing by linear factors. It involves using the coefficients of the polynomial and a specific value (the root of the divisor) to perform the division in a more efficient manner than traditional long division. This technique streamlines calculations and is especially helpful for higher-degree polynomials.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, the polynomial is x^4 - 256, which is a fourth-degree polynomial. Understanding the structure of polynomial functions is crucial for performing operations like division, as it helps identify the degree and behavior of the function.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is equal to f(c). This theorem is useful in synthetic division as it allows us to quickly find the remainder without completing the entire division process. It also provides insight into the roots of the polynomial and its graph.