Problem 16 - Introductory Algebra for College Students

Question

Multiply as indicated. $$\frac{3 y^{2}+17 y+10}{3 y^{2}-22 y-16} \cdot \frac{y^{2}-4 y-32}{y^{2}-8 y-48}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \( \frac{(y + 5)(y + 4)}{(y - 6)(y + 8)} \)

Step 1: Factorizing Polynomials

Start by factorizing each polynomial in the numerator and the denominator. This will make the problem easier to work with. \[ \frac{3y^2 + 17y + 10}{3y^2 - 22y - 16} × \frac{y^2 - 4y - 32}{y^2 - 8y - 48} \] factorizes to \[ \frac{(3y + 2)(y + 5)}{(3y + 2)(y - 8)} × \frac{(y - 8)(y + 4)}{(y - 6)(y + 8)} \]

Step 2: Cancelling Out Terms

Next, identify and cancel out the common terms that appear both in the numerator and the denominator. This helps in simplifying the expression. (3y + 2), (y - 8) can be cancelled out, thus the expression becomes \[ \frac{(y + 5)}{(1)} × \frac{(y + 4)}{(y - 6)(y + 8)} \]

Step 3: Multiplication

Multiply the fractions now that common terms have been cancelled out. This results in the simplified version of the expression. Floorman multiply the remaining terms, which results in the simplified version of the expression. \[ \frac{(y + 5)(y + 4)}{(y - 6)(y + 8)} \]