Divide using synthetic division. (x⁵+x³−2)/(x−1)

Verified step by step guidance

1. Identify the dividend and divisor. The dividend is the polynomial \(x^{5} + x^{3} - 2\) and the divisor is \(x - 1\).
2. Set up synthetic division by writing the coefficients of the dividend in descending order of powers. For \(x^{5} + x^{3} - 2\), the coefficients are \$1\( (for \)x^{5}\(), \)0\( (for \)x^{4}\(), \)1\( (for \)x^{3}\(), \)0\( (for \)x^{2}\(), \)0\( (for \)x\(), and \)-2$ (constant term).
3. Since the divisor is \(x - 1\), use \$1\( as the synthetic divisor (the zero of \)x - 1$). Write this number to the left of the coefficients.
4. Begin synthetic division: bring down the first coefficient (1) as is. Multiply it by the divisor (1) and write the result under the next coefficient. Add the column and write the sum below. Repeat this multiply-and-add process for all coefficients.
5. After completing the synthetic division, interpret the bottom row as the coefficients of the quotient polynomial, starting from one degree less than the original dividend, and the last number as the remainder.

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 and Missing Terms

When performing synthetic division, it is important to include coefficients for all powers of x, even if some terms are missing. For example, x^5 + x^3 - 2 should be written with zeros for x^4, x^2, and x terms to maintain proper alignment during division.

Remainder and Quotient Interpretation

The result of synthetic division gives a quotient polynomial and possibly a remainder. The quotient represents the division result without the remainder, and the remainder is the constant left over. The original polynomial equals (divisor × quotient) + remainder.