Perform each division. See Examples 9 and 10. (x^2+11x+16)/(x+8)

Verified step by step guidance

1. Identify the dividend \(x^2 + 11x + 16\) and the divisor \(x + 8\).
2. Set up the division by writing \(x^2 + 11x + 16\) under the long division symbol and \(x + 8\) outside.
3. Divide the first term of the dividend \(x^2\) by the first term of the divisor \(x\) to get the first term of the quotient, which is \(x\).
4. Multiply the entire divisor \(x + 8\) by this first term of the quotient \(x\) and subtract the result from the dividend.
5. Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. In this case, we divide the polynomial x^2 + 11x + 16 by the linear polynomial x + 8. The goal is to express the result as a quotient and possibly a remainder, which helps in simplifying expressions and solving equations.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For the polynomial x^2 + 11x + 16, we can factor it into (x + 8)(x + 2). This step is crucial because it can simplify the division process and help identify roots or solutions of the polynomial equation.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is f(c). This theorem is useful in polynomial division as it allows us to quickly find the remainder without performing the entire division process, providing insight into the behavior of the polynomial at specific values.