Problem 27 - Introductory Algebra for College Students

Question

Multiply as indicated. $$\frac{25-y^{2}}{y^{2}-2 y-35} \cdot \frac{y^{2}-8 y-20}{y^{2}-3 y-10}$$

Step-by-Step Solution

Answer

The result of the multiplication of the fractions given in the problem statement, after simplification, is \( \frac{(5-y)(y-10)}{(y-7)(y-5)} \).

Step 1: Factorize the Polynomials

Factorize every polynomial to its simplest form. The polynomials \(25-y^{2}\), \(y^{2}-2 y-35\), \(y^{2}-8 y-20\) , and \(y^{2}-3 y-10\) should be factorized as follows: \(25-y^{2} = (5+y)(5-y)\) , \(y^{2}-2 y-35 = (y-7)(y+5)\), \(y^{2}-8 y-20 = (y-10)(y+2)\) and \(y^{2}-3 y-10 = (y-5)(y+2)\).

Step 2: Insert the factorized polynomials into the fractions

Replace the polynomials in the original fractions with their factorized form: \(\frac{(5+y)(5-y)}{(y-7)(y+5)} \cdot \frac{(y-10)(y+2)}{(y-5)(y+2)}\). You should always make sure that you are not dividing by zero, which would make the function undefined.

Step 3: Simplify the fractions

Now, observe both fractions, looking for common factors in the numerators and denominators across the fractions. You should find that factors \( (y+2) \) and \( (y+5) \) are common and cancel these out. The simplified equation becomes: \( \frac{(5-y)(y-10)}{(y-7)(y-5)} \)