Problem 9 - Algebra 2 and Trigonometry

Question

In \(9-26,\) write each expression as the product of two binomials. $$ y(y+1)-1(y+1) $$

Step-by-Step Solution

Answer

The expression factors to \((y-1)(y+1)\).

Step 1: Identify Common Terms

Observe that the expression given is \( y(y+1) - 1(y+1) \). Notice that the term \((y+1)\) is common in both parts of the expression.

Step 2: Factor by Grouping

Since \((y+1)\) is common, you can factor \((y+1)\) out of the expression: \( y(y+1) - 1(y+1) = (y-1)(y+1) \).

Step 3: Verify the Factorization

To verify, expand \((y-1)(y+1)\) back to the original expression: \((y-1)(y+1) = y(y+1) - 1(y+1)\) using the distributive property. The factorization is correct.

Key Concepts

Binomial Multiplication

Multiplying binomials is an essential algebraic skill, particularly when working with polynomials. When dealing with the multiplication of two binomials, such as

• often this process involves expanding the expression through a method known as the FOIL technique:
  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms of the binomials.
  • Inner: Multiply the inner terms.
  • Last: Multiply the last terms in each binomial.
  
  Combining these results will give a polynomial, typically quadratic in nature. For example, consider expanding often seen when validating the result of binomial factorization like in This confirms the original expression after factorization.

Distributive Property

The distributive property is a fundamental rule in algebra that facilitates the distribution of a factor across addends inside parentheses. It's articulated as:

• This property holds that a single term multiplied by a sum inside parentheses can be rewritten as the sum of the individual products of each term from the sum.

In factorization, it helps to "un-collect" a term, allowing us to re-distribute them for simplification or verification. Consider the expression Here, distributing allows us to split the expression naturally. When we reverse this process, we effectively regroup them for factorization. This is what was used in the verification step discussed in the solution, ensuring that the expanded form reconstructs the original expression.

Common Factors

Identifying common factors is a strategic move to simplify expressions by finding terms that repeat within the polynomial. In the context of the problem, recognizing as a common factor permits us to "factor by grouping." Let's break it down:

• Find a GCF: Calculate a greatest common factor by identifying terms common to every part of the expression.
• Factor it out: Write the polynomial such that this common factor is outside a resulting pair of parentheses.
• Rewrite Simplified Form: This prepares the polynomial for further algebraic operations or simplifications.

In the exercise, acknowledging allowed for a simplification into easy-to-handle binomials that could be checked using reverse operations.