Use synthetic division to perform each division. (x3 - 1) / (x-1)

Ch. 3 - Polynomial and Rational Functions

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 21

Chapter 4, Problem 21

Use synthetic division to perform each division. (x³ - 1) / (x-1)

Verified step by step guidance

Identify the divisor and the dividend. The dividend is $x^3 - 1$ and the divisor is $x - 1$.

Set up synthetic division by writing the coefficients of the dividend. For $x^3 - 1$, the coefficients are $1$ (for $x^3$), $0$ (for $x^2$), $0$ (for $x$), and $-1$ (constant term).

Determine the value to use in synthetic division from the divisor $x - 1$. Since the divisor is $x - 1$, use $1$ (the root of $x - 1 = 0$) for synthetic division.

Perform synthetic division: bring down the first coefficient, multiply it by $1$, add to the next coefficient, and repeat this process across all coefficients.

Write the result of synthetic division as the quotient polynomial with one degree less than the dividend, and identify the remainder if any.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 Division

Polynomial division involves dividing one polynomial by another, similar to numerical division. Understanding how to divide polynomials helps in simplifying expressions, finding factors, and solving polynomial equations.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by x - c, the remainder is f(c). This concept helps verify the result of synthetic division and understand the relationship between division and function evaluation.

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