Problem 5 - Algebra and Trigonometry

Question

1–8 ? Factor out the common factor. $$ y(y-6)+9(y-6) $$

Step-by-Step Solution

Answer

Factor out (y-6) to get (y-6)(y+9).

Step 1: Identify the Common Factor

Examine each term in the expression \(y(y-6) + 9(y-6)\). Notice that both terms share a common factor of \((y-6)\).

Step 2: Rewrite the Expression

Rewrite the expression by factoring out the common term \((y-6)\) across both terms. This results in: \((y-6)(y+9)\).

Step 3: Confirm the Factored Expression

Verify the factored expression is correct by expanding it: \((y-6)(y+9) = y(y-6) + 9(y-6)\), which confirms the factoring was done correctly.

Key Concepts

common factor

A common factor is an algebraic term shared by two or more terms in an expression. Identifying a common factor is the foundational step in factoring. In the expression \( y(y-6)+9(y-6) \), the common factor is \( (y-6) \). Both terms, \( y(y-6) \) and \( 9(y-6) \), include this expression. Why is finding the common factor so important?

• It simplifies the expression by allowing you to "extract" the shared term.
• This process makes further algebraic manipulations, like solving equations, easier.

Think of it as finding the greatest number you can divide both parts by. Here, we're looking for an expression, not just a number. Once identified, you can use this common factor to simplify your algebraic work significantly.

factored expression

A factored expression results when you rewrite an original expression using its common factors. This process transforms a sum of terms into a product of terms. In our exercise, the expression \( y(y-6) + 9(y-6) \) is changed to \( (y-6)(y+9) \) after factoring. The benefit of having a factored expression is clear:

• It represents the expression in a more simplified and compact form.
• It reveals important characteristics, like roots or solutions, easier.

To confirm you've arrived at the correct factored form, you can always expand it back (covered in our next section) to ensure accuracy. Factored expressions are often more straightforward when solving equations and are essential for making many algebraic tasks manageable.

expand

Expanding is the reverse operation of factoring. It involves distributing terms to return a product back into a sum. In our example, if you take the factored expression \( (y-6)(y+9) \) and expand it, you end up back at the original expression \( y(y-6) + 9(y-6) \).Here’s how you do it:

• Apply the distributive property: multiply each term inside the first bracket by each term inside the second bracket.
• This yields \( y(y-6) + 9(y-6) \), confirming our factoring was correct.

Expanding is essential because it allows you to verify factored expressions. It ensures nothing was lost or changed in the factoring process. When you're sure both the original and the expanded forms match, you can confidently proceed with further problem-solving.