/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 Factor each polynomial. $$ \te... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 36

Factor each polynomial. $$ \text { 36. }(m-n)^{2}+4(m-n)+4 $$

Short Answer

Expert verified
((m-n) + 2)^2 or (m-n+2)^2.

Step by step solution

01

Recognize the quadratic form

Notice that the given polynomial \( (m-n)^{2} + 4(m-n) + 4 \) is in the form of a quadratic expression where \( m-n \) is the variable. Let's write it as \( x^{2} + 4x + 4 \) with \( x = (m-n) \).
02

Identify the quadratic terms

In the expression \( x^{2} + 4x + 4 \), identify the coefficients of the quadratic, linear, and constant terms: \( a = 1 \), \( b = 4 \), and \( c = 4 \).
03

Factor the quadratic expression

To factor \( x^{2} + 4x + 4 \), find two numbers that multiply to \( c \) (which is 4) and add up to \( b \) (which is 4). The numbers 2 and 2 satisfy these conditions because \( 2 \times 2 = 4 \) and \( 2 + 2 = 4 \). Therefore, \( x^{2} + 4x + 4 \) can be factored as \( (x + 2)(x + 2) \) or \( (x + 2)^{2} \).
04

Substitute back the original variable

Replace \( x \) with \( (m-n) \) in the factored expression to get the final factored form. The expression becomes \( ((m-n) + 2)^{2} \). Simplifying inside the parentheses, we obtain \( (m-n+2)^{2} \).

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Key Concepts

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

Quadratic Expressions
Quadratic expressions are polynomial expressions of degree two. This means that the highest power of the variable in the expression is two. Quadartic expressions typically take the form \[ ax^{2} + bx + c \]where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Notice in the original problem, the expression \( (m-n)^{2} + 4(m-n) + 4 \) is essentially a quadratic expression with \( m-n \) acting as the variable.
Factoring Techniques
Factoring is the process of breaking down a complex expression into simpler components whose product is the original expression. When factoring quadratic expressions, one common technique is to find two numbers that:
  • Multiply to give the constant term \( c \)
  • Add up to give the coefficient of the linear term \( b \).
For example, in the step-by-step solution provided, the quadratic expression \( x^{2} + 4x + 4 \) requires finding two numbers that multiply to 4 and add up to 4. These numbers are 2 and 2. Hence, we can factor the expression as \((x + 2)(x + 2)\) or \((x + 2)^{2}\).
Algebraic Identities
Algebraic identities are equations that hold true for any value of the variable. Knowing these identities can simplify the factoring process. One very useful identity is the perfect square trinomial: \[ (a + b)^{2} = a^{2} + 2ab + b^{2} \].In the given problem, the expression \((m-n)^{2} + 4(m-n) + 4\) can be recognized as a perfect square trinomial. By rewriting it as \((m-n + 2)^{2}\), the factoring becomes more straightforward. This makes the process of recognizing patterns in algebraic expressions much more efficient.

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