Use synthetic division to perform each division. x^4-1 / x-1

Verified step by step guidance

1. Identify the divisor and the dividend. Here, the divisor is \(x - 1\) and the dividend is \(x^4 - 1\).
2. Set up synthetic division by writing the coefficients of the dividend. For \(x^4 - 1\), the coefficients are \([1, 0, 0, 0, -1]\).
3. Use the root of the divisor \(x - 1\), which is \(1\), as the number to use in synthetic division.
4. Bring down the leading coefficient \(1\) to the bottom row.
5. Multiply the root \(1\) by the number just written on the bottom row, and write the result under the next coefficient. Add this result to the next coefficient and write the sum below the line. Repeat this process for all coefficients.

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 a specific value (c) to perform the division without writing out the entire polynomial long division. This technique is particularly useful for quickly finding the quotient and remainder when dividing by linear factors.

Polynomials

A polynomial is a mathematical expression consisting of variables raised to non-negative integer powers and coefficients. In the context of the question, x^4 - 1 is a polynomial of degree 4, which can be factored or divided by another polynomial. Understanding the structure of polynomials is essential for applying synthetic division effectively.

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 is useful in synthetic division as it allows us to quickly determine the remainder without fully performing the division. It also helps in understanding the relationship between the roots of the polynomial and its factors.