Perform each division. See Examples 9 and 10. (x²+11x+16)/(x+8)

Verified step by step guidance

1. Identify the division problem as a polynomial division: divide the quadratic polynomial \(x^2 + 11x + 16\) by the linear polynomial \(x + 8\).
2. Set up the long division by writing \(x^2 + 11x + 16\) under the division bar and \(x + 8\) outside the division bar.
3. Divide the leading term of the dividend \(x^2\) by the leading term of the divisor \(x\) to get the first term of the quotient: \(\frac{x^2}{x} = x\).
4. Multiply the entire divisor \(x + 8\) by this term \(x\) and subtract the result from the dividend to find the new remainder.
5. Repeat the process with the new remainder: divide the leading term of the remainder by \(x\), multiply the divisor by this term, subtract again, and continue until the degree of the remainder is less than the degree of the divisor.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree, similar to numerical long division. It involves dividing the leading terms, multiplying, subtracting, and bringing down the next term until the remainder is of lower degree than the divisor.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. Recognizing factorable expressions can simplify division problems, especially when the divisor is a binomial, allowing for easier simplification or verification of the quotient.

Remainder and Quotient Interpretation

When dividing polynomials, the result includes a quotient and possibly a remainder. Understanding how to express the division result as quotient plus remainder over divisor helps interpret the final answer correctly, especially if the division is not exact.