Use synthetic division to perform each division. (x⁴ + 4x³ + 2x² + 9x+4) / x+4

Verified step by step guidance

1. Identify the divisor and rewrite it in the form \(x - c\). Since the divisor is \(x + 4\), rewrite it as \(x - (-4)\), so \(c = -4\).
2. Write down the coefficients of the dividend polynomial \(x^4 + 4x^3 + 2x^2 + 9x + 4\). These are: 1, 4, 2, 9, and 4.
3. Set up the synthetic division tableau by placing \(c = -4\) to the left and the coefficients in a row: 1, 4, 2, 9, 4.
4. Perform synthetic division by bringing down the first coefficient (1), then multiply it by \(c\) (-4), add the result to the next coefficient, and repeat this process across all coefficients.
5. Interpret the final row of numbers: the first four numbers represent the coefficients of the quotient polynomial (starting from degree 3), and the last number is 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 and performing arithmetic operations in a tabular form. This method is efficient for finding quotients and remainders quickly.

Polynomial Coefficients and Setup

To use synthetic division, you must identify and list the coefficients of the dividend polynomial in descending order of degree. If any terms are missing, their coefficients are represented as zero. Proper setup ensures accurate calculations during the synthetic division process.

Interpreting the Divisor in Synthetic Division

In synthetic division, the divisor x + 4 is rewritten as x - (-4), so the value used in the process is -4. Recognizing this sign change is crucial because synthetic division uses the root of the divisor (the value that makes it zero) to perform the calculations correctly.