/*! 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 116 Explain why \(a^{2}+2 a+1\) is a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 116

Explain why \(a^{2}+2 a+1\) is a perfect-square trinomial and why \(a^{2}+4 a+1\) isn't a perfect-square trinomial.

Short Answer

Expert verified
\(a^2 + 2a + 1\) is the square of \((a+1)\), while \(a^2 + 4a + 1\) does not fit the perfect-square pattern.

Step by step solution

01

Understand a Perfect-Square Trinomial

A perfect-square trinomial is an expression in the form \((a+b)^2 = a^2 + 2ab + b^2\). It expands to a square of a binomial. We will test each given trinomial against this form.
02

Rewrite and Identify Structure for \(a^2 + 2a + 1\)

Compare \(a^2 + 2a + 1\) to \((a+b)^2 = a^2 + 2ab + b^2\). Here, we can see that \(a^2\) corresponds to \(a^2\), and \(1\) corresponds to \(b^2\), where \(b = 1\). The middle term, \(2a\), is \(2ab\), which means \(2a = 2 \cdot a \cdot 1\). This confirms \(b = 1\) and the trinomial represents \((a+1)^2\).
03

Verify if \(a^2 + 2a + 1\) is a Perfect-Square

Check if \(a^2 + 2a + 1\) can be rewritten as \((a+1)^2\). Expanding \((a+1)^2\) gives \(a^2 + 2a + 1\), confirming that this expression is indeed a perfect-square trinomial.
04

Check Structure for \(a^2 + 4a + 1\)

Now, compare \(a^2 + 4a + 1\) to \((a+b)^2 = a^2 + 2ab + b^2\). We attempt to match \(a^2 + 4a + 1\) with this structure. Here, \(4a\) must equal \(2ab\). If we solve for \(b\), setting \(4a = 2 \cdot a \cdot b\), we find \(b = 2\). But \(b^2\) would then be \(4\), not \(1\), thus it does not fit the pattern of a perfect-square.
05

Conclusion on \(a^2 + 4a + 1\)

Since we cannot match \(a^2 + 4a + 1\) to the form \((a+b)^2\) due to a discrepancy in the constant term, \(a^2 + 4a + 1\) is not a perfect-square trinomial.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Algebraic Expressions
Algebraic expressions are a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They represent a value and are foundational in algebra. Understanding these expressions is crucial for solving problems, especially when they form equations or more complex structures like polynomials.

Expressions are made up of:
  • **Terms**: The parts of an expression separated by plus or minus signs. In \(a^2 + 2a + 1\), you have three terms: \(a^2\), \(2a\), and \(1\).
  • **Coefficients**: Numbers multiplying variables. In \(2a\), \(2\) is the coefficient.
  • **Constants**: Numbers without variables, like \(1\) in the expression \(a^2 + 2a + 1\).
Grasping algebraic expressions is essential, as it helps in simplifying and manipulating expressions to solve equations and factor polynomials efficiently.
Factoring Trinomials
Factoring trinomials is a process where you express a trinomial as the product of two binomials. Recognizing and factoring a perfect-square trinomial is a common task. A perfect-square trinomial can be expressed in the form \((a+b)^2 = a^2 + 2ab + b^2\).

Let's explore:
  • Compare \(a^2 + 2a + 1\) to \((a+b)^2\): Notice \(a^2\) and \(1\) align with \(a^2\) and \(b^2\), where \(b=1\). The middle term \(2a\) confirms \(b=1\) because it completes \(2ab = 2 \cdot a \cdot 1\).
  • Write it as \((a+1)^2\): By verifying through expansion \((a+1)^2\), get back the original trinomial, confirming it's a perfect-square.
  • Challenge in \(a^2 + 4a + 1\): When attempting to match with \((a+b)^2\), the middle term requires \(b = 2\), but this changes the constant term from \(1\) to \(4\), proving it isn't a perfect-square trinomial.
By working through these patterns, you quickly learn to factor correct forms and address mismatches.
Square of a Binomial
The square of a binomial is a vital concept in algebra, helping simplify and solve expressions. The expression is expanded as \((a+b)^2 = a^2 + 2ab + b^2\).

Consideration of binomial squares include:
  • Recognizing form: To see if a trinomial like \(a^2 + 2a + 1\) is a square of a binomial, compare each component with the general expanded form. If you can align all terms with \(a^2\), \(2ab\), and \(b^2\), it's a match.
  • Multiplying out binomials: Also refer to this as expanding. When expanding \((a+1)^2\), it results in \(a^2 + 2a + 1\), perfectly matching the original trinomial.
  • Limits of pattern: Not all trinomials fit in \((a+b)^2\). Expression \(a^2 + 4a + 1\) fails this test as the middle term demands a different constant term than what emerges when squaring a binomial.
Understanding the form helps in factoring, expanding, and transforming expressions efficiently鈥攌ey skills in algebraic problem-solving.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A bag of peanuts is worth \(\$ 0.30\) less than the same size bag of cashews. Equal amounts of peanuts and cashews are used to make 40 bags of a mixture that is worth \(\$ 1.05\) per bag. How much is a bag of cashews worth?

For each of the following polynomials, which factoring method would you use first? $$ x^{3}+27 $$

Evaluate each expression. $$ \frac{1}{7^{-2}} $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 4 t^{2}+36 $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 16 d^{2}-56 d z+49 z^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.