Problem 71 - Introductory Algebra for College Students

Question

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y-3)(y+2)}{(y+1)(y-4)}-\frac{(y+2)(y+3)}{(y+1)(4-y)}-\frac{(y+5)(y-1)}{(y+1)(4-y)}$$

Step-by-Step Solution

Answer

The simplified version of the given expression is \(\frac{3y^2+8y-17}{(y+1)(y-4)}\).

Step 1 Distribute terms in the numerator

Distribute the terms within the parenthesis in the numerators of the factions \((y-3)(y+2)\), \((y+2)(y+3)\), \((y+5)(y-1)\). Get \(y^2-y-6\), \(y^2+5y+6\) and \(y^2+4y-5\) respectively.

Step 2 Notice the similarity of the denominators

Observe that the denominator \((y+1)(4-y)\) is same as \((y+1)(y-4)\) but with a negative sign because \(4-y=-(y-4)\)

Step 3 Combine fractions based on similar denominators

Combine the fractions that have the same denominator. The operands are subtracted. We end up with\[\frac{(y^2-y-6)-(y^2+5y+6)-(y^2+4y-5)}{(y+1)(y-4)}\]

Step 4 Combine like terms in the numerator

Group the similar terms in the numerator. We have \[\frac{3y^2+8y-17}{(y+1)(y-4)}\]

Step 5 Final Solution

The simplified version of the given problem is \[\frac{3y^2+8y-17}{(y+1)(y-4)}\]

Key Concepts

Distributing Terms

Understanding how to distribute terms across parentheses is essential when working with algebraic expressions. This process involves multiplying each term within one set of parentheses by every term in the other set. This step provides the groundwork for simplification. For instance, take the expression \(y-3)(y+2)\). By applying the distributive property, we multiply \((y \times y) + (y \times 2) - (3 \times y) - (3 \times 2)\), which simplifies to \((y^2 + 2y - 3y - 6)\).

After distributing the terms, we collect like terms to make the expression more concise. Using distribution helps transform the complicated fraction into a more manageable form, allowing for further simplification. It’s vital to perform this step correctly, as any mistake will carry through the rest of the problem, leading to incorrect solutions.

Combining Like Terms

Once terms are distributed, the next step is to combine like terms. Like terms are mathematical terms that have the same variable raised to the same power. In our example, after distributing terms through multiplication, we end with expressions such as \(y^2 - y - 6\), which cannot be simplified further. Yet, when these expressions come together in a larger algebraic fraction, we align like terms to combine them.

During the combination, we add or subtract coefficients of like terms. In our exercise, after subtracting the distributed terms, we ended up with \(3y^2 + 8y - 17\) in the numerator. This step is a neat consolidation that makes the expression cleaner and sets the stage for any subsequent operations, such as factoring or, in this case, arriving at the final simplified form of the algebraic fraction.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying them often requires the steps we’ve discussed: distributing and combining like terms. In our exercise, after amalgamating terms, we saw that the denominators of the fractions were the same, albeit one was the negative of the other. Recognizing this similarity is critical.

When we have a consistent denominator across multiple fractions, it allows us to combine the numerators directly, ensuring we do not violate any mathematical rules. The denominator's behavior—the flipping of signs due to the negative—highlights the importance of paying attention to sign changes, which can alter the expression's entire structure. The final simplified rational expression, \(\frac{3y^2 + 8y - 17}{(y + 1)(y - 4)}\), is the result of careful execution of these algebraic maneuvers.