Perform each division. See Examples 9 and 10. (q²+4q-32)/(q-4)

Verified step by step guidance

1. Identify the division problem as a polynomial division: \(\frac{q^2 + 4q - 32}{q - 4}\).
2. Set up the division using either long division or synthetic division. For long division, write \(q^2 + 4q - 32\) under the division bar and \(q - 4\) outside.
3. Divide the leading term of the numerator \(q^2\) by the leading term of the denominator \(q\) to get the first term of the quotient: \(q\).
4. Multiply the entire divisor \(q - 4\) by this term \(q\) and subtract the result from the original polynomial to find the new remainder.
5. Repeat the process with the new remainder: divide the leading term of the remainder by \(q\), 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 Division

Polynomial division is the process of dividing one polynomial by another, similar to long division with numbers. It helps simplify expressions or find quotients and remainders. In this problem, dividing (q^2 + 4q - 32) by (q - 4) involves determining how many times the divisor fits into the dividend.

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials. Recognizing factors can simplify division problems by canceling common terms. For example, factoring q^2 + 4q - 32 may reveal a factor of (q - 4), making division straightforward.

Remainder Theorem

The Remainder Theorem states that the remainder of dividing a polynomial f(x) by (x - c) is f(c). This helps verify the result of polynomial division and determine if the divisor is a factor. Evaluating the polynomial at q = 4 can confirm if (q - 4) divides evenly.