In Exercises 17–32, divide using synthetic division. (4x^3−3x^2+3x−1)÷(x−1)

Verified step by step guidance

1. Identify the divisor and ensure it is in the form (x - c). Here, the divisor is (x - 1), so c = 1.
2. Write down the coefficients of the dividend polynomial 4x^3 - 3x^2 + 3x - 1, which are 4, -3, 3, and -1.
3. Set up the synthetic division by writing c = 1 to the left and the coefficients 4, -3, 3, -1 in a row.
4. Bring down the leading coefficient (4) to the bottom row.
5. Multiply c (which is 1) by the number just written on the bottom row (4), and write the result under the next coefficient (-3). Add this result to -3 and write the sum below. Repeat this process for the remaining coefficients.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing polynomials, particularly useful when dividing by linear factors of the form (x - c). It involves using the coefficients of the polynomial and a specific value (c) to perform the division in a more efficient manner than traditional long division.

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 4x^3 - 3x^2 + 3x - 1 is a cubic polynomial, which means its highest degree is three, indicating it can have up to three roots or solutions.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (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 value of the polynomial at a specific point, which can help in verifying the results of the division.