In Exercises 17–32, divide using synthetic division. (4x³−3x²+3x−1)÷(x−1)

Verified step by step guidance

1. Identify the dividend and divisor. The dividend is the polynomial $\mathrm{4x^3}-\mathrm{3x^2}+\mathrm{3x}-1$ and the divisor is $x\; -\; 1$.
2. Set up synthetic division by writing the coefficients of the dividend in order: $4,\; -3,\; 3,\; -1$. Since the divisor is $x\; -\; 1$, use $1$ as the synthetic divisor (the zero of $x\; -\; 1$).
3. Draw the synthetic division setup: write the number 1 to the left, and the coefficients $4,\; -3,\; 3,\; -1$ in a row to the right.
4. Perform synthetic division by bringing down the first coefficient (4), then multiply it by 1 and add to the next coefficient: $4\; imes\; 1\; +\; (-3)\; =\; 1$. Repeat this process for all coefficients.
5. The numbers obtained after the last addition represent the coefficients of the quotient polynomial, and the final number is the remainder. Write the quotient polynomial using these coefficients with decreasing powers of $x$ starting from one degree less than the dividend.

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.

Polynomial Coefficients

Polynomial coefficients are the numerical factors in front of the variable terms. In synthetic division, these coefficients are arranged in descending order of degree and used as the main elements for the division process.

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 concept helps verify the result of synthetic division and understand the relationship between division and evaluation of polynomials.