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Chapter 5: Problem 74

Use a substitution to help factor each expression. See Example 10. $$ (a+b)^{2}-2(a+b)+1 $$

Short Answer

Expert verified
The expression \((a+b)^2 - 2(a+b) + 1\) factors into \((a+b-1)^2\).

Step by step solution

01

Recognize Substitution Opportunity

First, we notice that the expression \((a+b)^2 - 2(a+b) + 1\) has a repeated term \(a+b\). This suggests that a substitution can simplify the expression. Let's set \(u = a + b\). This simplifies our problem to: \(u^2 - 2u + 1\).
02

Substitute and Simplify

Substitute \(u = a + b\) into the expression, which gives us \(u^2 - 2u + 1\). This new expression is in terms of a single variable \(u\).
03

Recognize the Perfect Square

The expression \(u^2 - 2u + 1\) resembles a perfect square trinomial, which can be factored as \((u-1)^2\).
04

Substitute Back

Now substitute back the original expression of \(u\), which is \(a+b\). Therefore, \((u-1)^2 = ((a+b)-1)^2 = (a+b-1)^2\).
05

Final Factored Form

The original expression \((a+b)^2 - 2(a+b) + 1\) can now be expressed as \((a+b-1)^2\), which is its fully factored form.

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique used to simplify complex algebraic expressions by temporarily replacing part of the expression with a single variable. The idea is to create a simpler form that is easier to work with and then revert back to the original terms once the simplification is complete.

  • Identify Repeated Terms: Look for terms that appear more than once in the expression. These are good candidates for substitution.
  • Assign a Variable: Replace the repeated term with a single variable, often denoted as "u" to simplify the expression.
  • Simplify: Operate on the simpler expression to find an easier solution or form.
  • Re-substitute: Once simplified, replace the variable back with the original terms to find your solution.
In the context of our exercise, we recognized that \(a+b\) appeared multiple times, allowing us to use the substitution \(u = a + b\). This simplified the expression to \(u^2 - 2u + 1\), making it more straightforward to factor.
Perfect Square Trinomial
A perfect square trinomial is a special form of a quadratic expression that can be written as the square of a binomial. Recognizing perfect squares is the key to factoring these trinomials effectively.

A trinomial with the form \((ax)^2 + 2abx + b^2\) or \((ax)^2 - 2abx + b^2\) can be factored as \((ax + b)^2\) or \((ax - b)^2\), respectively. Here's a breakdown:

  • Standard Form: Look for expressions that fit the form \(x^2 \pm 2xy + y^2\), which are recognizable perfect squares.
  • Factorization: The expression can be directly rewritten as the square of a binomial.
For example, in our rephrased expression \(u^2 - 2u + 1\), it matches the pattern of a perfect square \((u-1)^2\). Understanding this pattern makes it significantly easier to factor.
Algebraic Expressions
Algebraic expressions represent a combination of numbers, variables, and operational symbols. They are a fundamental aspect of algebra and understanding how to manipulate these is crucial for problem-solving.

  • Components: Elements like constants, variables, and coefficients come together to form expressions.
  • Operations: Expressions can involve addition, subtraction, multiplication, division, and exponentiation.
  • Simplification and Factoring: These processes help in making expressions easier to work with, by reducing them to simpler forms or finding equivalent expressions.
Our original exercise involves manipulating the algebraic expression \((a+b)^2 - 2(a+b) + 1\)\, which required comprehension of the relationship between its terms and variables. Through substitution and recognizing patterns like the perfect square trinomial, we factored it into a simpler form \((a+b-1)^2\). This exercise highlights the importance of algebraic manipulation in solving complex expressions.

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Most popular questions from this chapter

Multiply. Assume \(n\) is a natural number. $$ \left(a^{3 n}-b^{3 n}\right)\left(a^{3 n}+b^{3 n}\right) $$

Explain why the factorization is not complete. $$ \text { Factor: } \quad 1-t^{8}=\left(1+t^{4}\right)\left(1-t^{4}\right) $$

What polynomial should be subtracted from \(5 x^{3} y-5 x y+2 x\) to obtain the polynomial \(8 x^{3} y-7 x y+11 x ?\)

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