Perform each division. See Examples 9 and 10. (q^2+4q-32)/(q-4)

Verified step by step guidance

1. Identify the dividend \( q^2 + 4q - 32 \) and the divisor \( q - 4 \).
2. Set up the division by writing \( q^2 + 4q - 32 \) under the long division symbol and \( q - 4 \) outside.
3. Divide the first term of the dividend \( q^2 \) by the first term of the divisor \( q \) to get the first term of the quotient, which is \( q \).
4. Multiply the entire divisor \( q - 4 \) by this first term of the quotient \( q \) 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. It involves determining how many times the divisor can fit into the leading term of the dividend, subtracting the result, and repeating the process with the remainder until the degree of the remainder is less than that of the divisor.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler components, or factors. This is crucial in simplifying expressions and solving equations, as it can reveal roots or zeros of the polynomial. For example, the polynomial q^2 + 4q - 32 can be factored to facilitate division.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(q) is divided by a linear divisor of the form (q - c), the remainder of this division is equal to f(c). This theorem is useful for quickly finding the remainder without performing full polynomial long division, and it can help in evaluating the division of polynomials.