Divide using synthetic division. (x⁷+x⁵−10x³+12)/(x+2)

Verified step by step guidance

1. Identify the divisor and rewrite it in the form \( x - c \). Since the divisor is \( x + 2 \), rewrite it as \( x - (-2) \), so \( c = -2 \).
2. Write down the coefficients of the dividend polynomial \( x^7 + x^5 - 10x^3 + 12 \). Include zeros for any missing powers of \( x \). The coefficients are: \( 1, 0, 1, 0, -10, 0, 0, 12 \) corresponding to \( x^7, x^6, x^5, x^4, x^3, x^2, x^1, x^0 \).
3. Set up the synthetic division by writing \( c = -2 \) to the left and the coefficients in a row to the right.
4. Begin the synthetic division process: bring down the first coefficient, multiply it by \( c \), add to the next coefficient, and repeat this process across all coefficients.
5. The final row of numbers (except the last one) will be the coefficients of the quotient polynomial, and the last number will be 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. Missing terms should be represented by zero coefficients to maintain the correct alignment during division.

Interpreting the Divisor in Synthetic Division

In synthetic division, the divisor x + 2 is rewritten as x - (-2), so the value used in the synthetic division process is -2. Recognizing this sign change is crucial for correctly setting up and performing the division.