Polynomial Division with Ruffini's Rule

Overview

This lecture explains how to divide polynomials using the Ruffini rule, including when it applies, step-by-step examples, and key points for correctly setting up and solving such divisions.

When to Use the Ruffini Rule

• The Ruffini rule is used only when the divisor is a binomial of the form x ± a, where a is any number.
• For other types of divisors, use the full polynomial division method.

Setting Up for Ruffini Division

• Arrange the dividend polynomial in descending order of exponents.
• Write coefficients for each term, inserting zeros for any missing degree terms.
• The divisor x ± a is set equal to zero to find the "root" (solution for x), which is used in Ruffini's method.

Steps in Ruffini Division

• Write the coefficients in a row.
• Place the root (solution for x in the divisor) to the left of the row.
• Drop down the first coefficient.
• Multiply the dropped number by the root and write the result under the next coefficient.
• Add vertically and continue multiplying and adding for all coefficients.
• The last number is the remainder; the rest give the coefficients of the quotient polynomial.

Examples and Key Points

• Always insert a zero for any missing degree terms in the dividend.
• When the dividend's degree is n, the quotient's degree will be n – 1.
• Apply the procedure the same way for fractions as divisors or coefficients.

Key Terms & Definitions

• Ruffini rule — a simplified method for dividing polynomials when the divisor is x ± a.
• Dividend — the polynomial being divided.
• Divisor — the binomial x ± a used to divide the dividend.
• Root — the value of x that makes the divisor zero.
• Coefficient — the numerical factor of each term in a polynomial.
• Remainder — the last number in Ruffini's process, representing any leftover value after division.
• Quotient — the resultant polynomial after division.

Action Items / Next Steps

• Practice Ruffini division with polynomials, ensuring correct arrangement and handling of missing terms.
• Review the video on full polynomial division if unsure about dividing by non-binomial divisors.