Use synthetic division to perform each division. x⁴-1 / 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 zero of the divisor \(x - 1 = 0\), which is \(x = 1\). Then list the coefficients of the dividend polynomial in descending order of degree. For \(x^4 - 1\), the coefficients are \$1\( (for \)x^4\(), \)0\( (for \)x^3\(), \)0\( (for \)x^2\(), \)0\( (for \)x\(), and \)-1$ (constant term).
3. Begin synthetic division by bringing down the first coefficient (which is 1) as is. Then multiply this number by the zero of the divisor (1), and write the result under the next coefficient.
4. Add the column values and write the sum below. Repeat the multiply and add process for each coefficient until all coefficients have been processed.
5. The numbers obtained at the bottom row (except the last one) represent the coefficients of the quotient polynomial. The last number is the remainder. Write the quotient polynomial using these coefficients with descending powers of \(x\) starting from one degree less than the original polynomial.

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

Understanding polynomial coefficients is essential because synthetic division operates on these numerical values. Each term's coefficient is used in the division process, and missing terms must be represented by zero coefficients to maintain the correct order.

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 polynomial evaluation.