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Chapter 1: Problem 9

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

Short Answer

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

Step by step solution

01

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.
02

Factor by Grouping

Since \((y+1)\) is common, you can factor \((y+1)\) out of the expression: \( y(y+1) - 1(y+1) = (y-1)(y+1) \).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In \(13-22,\) solve each equation or inequality. Each solution is an integer. $$ 2 x+3 < x+15 $$

In \(9-26,\) write each expression as the product of two binomials. $$ 9 x^{2}-12 x+4 $$

In \(27-39,\) factor each polynomial completely. $$ (c+2)^{2}-1 $$

In \(3-17,\) solve each equation or inequality. Each solution is an integer. $$ |3-y|=8 $$

In \(3-12,\) write the sum or difference of the given polynomials in simplest form. $$ \left(2 a^{4}-5 a^{2}-1\right)+\left(a^{3}+a\right) $$
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