Welcome back to Math 11! Today, we’re exploring the exciting world of multiplying fractions that include binomials with variables. This skill is essential for solving more complex algebraic problems, and it builds on your understanding of basic fraction multiplication. Let’s dive right in!

Review: Multiplying Fractions

First, let’s review how to multiply simple fractions. When multiplying two fractions, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together:

Multiplying Fractions with Binomials and Variables

Now, let’s add binomials with variables into the mix. Multiplying fractions with binomials follows the same principles as multiplying numerical fractions. Here’s how it works:

1. Multiply the Numerators: Multiply the expressions in the numerators together.
2. Multiply the Denominators: Multiply the expressions in the denominators together.
3. Simplify: Factor out any common factors in the numerator and denominator, and simplify the fraction if possible.

Examples: Multiplying Fractions with Binomials and Variables

Example 1:

1. • Multiply the numerators: (x+2)×3=3(x+2)
   • Multiply the denominators: (x−3)×(x+4)=(x−3)(x+4)
   • Result: 3(x+2)/(x−3)(x+4)
2. Example 2:

1. • Factor 2x+10 in the numerator: 2(x+5)
   • Multiply the numerators: (x−1)×2(x+5)=2(x−1)(x+5)
   • Multiply the denominators: (x+5)×(x−2)=(x+5)(x−2)
   • Simplify by canceling common factors: x+5
   • Result: 2(x−1)/x−2
2. Example 3:

1. • Factor x-4^2 in the numerator: (x−2)(x+2)
   • Multiply the numerators: (x−2)(x+2)×(x+3)=(x−2)(x+2)(x+3)
   • Multiply the denominators: (x+3)×(x−2)=(x+3)(x−2)
   • Simplify by canceling common factors: x+3 and x−2
   • Result: x+2
2. Example 4:

1. • Factor 2x+6in the numerator: $2(x+3)$
   • Factor x−9^2in the denominator: (x+3)(x−3)
   • Multiply the numerators: 2(x+3)×(x+3)=2(x+3)^2
   • Multiply the denominators: (x+3)(x−3)×4x=4x(x+3)(x−3)
   • Simplify by canceling common factors: x+3

Conclusion

Multiplying fractions with binomials and variables is a valuable skill that extends your understanding of basic fraction multiplication. By following the steps of multiplying numerators and denominators and then simplifying, you can tackle complex algebraic expressions with confidence. Keep practicing these techniques, and soon you’ll master the art of multiplying fractions with binomials and variables!