Multiply or divide, as indicated. See Example 3. 4a + 12 a² - 9 ————— ÷ ————— 2a - 10 a² - a - 20

Verified step by step guidance

1. Step 1: Factor the numerator $4a+12a^2-9$ and the denominator $2a-10$ of the first fraction.
2. Step 2: Factor the numerator $a^2-a-20$ of the second fraction.
3. Step 3: Rewrite the division of fractions as multiplication by the reciprocal of the second fraction.
4. Step 4: Simplify the expression by canceling out common factors in the numerator and the denominator.
5. Step 5: Combine the remaining factors to express the simplified result.

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. Understanding this concept is crucial for simplifying expressions and solving equations involving polynomials.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler components, or factors. This is essential for simplifying expressions and performing polynomial division, as it allows for cancellation of common factors. Recognizing patterns such as the difference of squares or grouping can aid in this process.

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Understanding how to manipulate these expressions, including multiplying, dividing, and simplifying, is vital in algebra and trigonometry. This concept also includes recognizing restrictions on variable values to avoid division by zero.