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Chapter 6: Problem 27

Expand each expression. Simplify your expansion if possible. \((4 x+1)^{2}\)

Short Answer

Expert verified
The expansion of \((4x + 1)^2\) is \16x^2 + 8x + 1\.

Step by step solution

01

- Understand the Formula

Recognize that the problem requires expanding \((4x + 1)^2\). Use the algebraic identity for the square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\).
02

- Identify 'a' and 'b'

In this case, \(a = 4x\) and \(b = 1\).
03

- Apply the Formula

Substitute \(a\) and \(b\) into the formula: \( (4x + 1)^2 = (4x)^2 + 2(4x)(1) + 1^2 \).
04

- Perform Calculations

Calculate each term: \((4x)^2 = 16x^2\), \(2(4x)(1) = 8x\), and \(1^2 = 1\).
05

- Combine the Terms

Add all the terms together: \ 16x^2 + 8x + 1 \Therefore, the expanded and simplified form of \( (4x + 1)^2 \) is \ 16x^2 + 8x + 1 \.

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Key Concepts

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

binomial theorem
The binomial theorem is a fundamental concept for expanding expressions raised to a power. Specifically, it helps us find the expanded form of \((a+b)^n\). In simpler terms, this means when you have a binomial鈥攖wo terms, a and b鈥攁dded and raised to a power, there is a specific way to expand it. For example, the binomial theorem tells us: \[ (a+b)^2 = a^2 + 2ab + b^2 \] This is why we use the pattern: one term squared, two times the product of both terms, and the second term squared. For higher powers, the theorem involves more terms, but the concept remains consistent.
polynomial expansion
Polynomial expansion is simply applying the rules like the binomial theorem to stretch a compact polynomial into its expanded form. For instance, the quadratic expression \((4x+1)^2\) is a polynomial that can be expanded using our algebraic patterns. The steps are:
  • First term squared: \((4x)^2 = 16x^2\)
  • Two times product of both terms: \(2(4x)(1) = 8x\)
  • Second term squared: \(1^2 = 1\)
Combine these terms together and you get the expanded form: \[16x^2 + 8x + 1\]
algebraic identities
Algebraic identities are like rules or formulas that are always true. They help simplify and solve algebraic expressions easily. For instance, one common identity is the square of a binomial: \( (a + b)^2 = a^2 + 2ab + b^2 \) This identity helps expand the expression \((4x+1)^2\). By recognizing these patterns, we can quickly substitute \(a=4x\) and \(b=1\) into the identity, making our work clearer and reducing errors. Knowing a few key identities saves time and effort in algebra.
simplifying expressions
Simplifying expressions means making them easier to understand or solve. After expanding a polynomial, simplify it by combining like terms. For example, with \((4x + 1)^2\), we expanded to get \16x^2 + 8x + 1\. Since there are no like terms to combine further, this is our simplified form.
Simplifying involves:
  • Breaking down complex expressions
  • Performing arithmetic on like terms
In algebra, the goal is always to make the expression as neat and readable as possible.

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Most popular questions from this chapter

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