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Chapter 1: Problem 41

Perform the indicated multiplications. $$(3 x-7)^{2}$$

Short Answer

Expert verified
The result of \((3x-7)^2\) is \(9x^2 - 42x + 49\).

Step by step solution

01

Understand the Problem

We need to expand the expression \((3x - 7)^2\). This means we are multiplying \((3x - 7)\) by itself.
02

Apply the Formula for Squaring a Binomial

The formula for squaring a binomial \((a - b)^2\) is \(a^2 - 2ab + b^2\). Here, \(a = 3x\) and \(b = 7\).
03

Calculate Each Part of the Formula

First, calculate \(a^2 = (3x)^2 = 9x^2\). Next, find \(2ab = 2(3x)(7) = 42x\). Finally, calculate \(b^2 = (7)^2 = 49\).
04

Substitute Back into the Formula

Replace \(a^2\), \(2ab\), and \(b^2\) into the formula: \((3x - 7)^2 = 9x^2 - 42x + 49\).

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Key Concepts

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

Polynomial Multiplication
Polynomial multiplication is a fundamental concept in algebra that students often encounter when expanding binomials. It's the process of multiplying two or more polynomials together to form a single polynomial. In our exercise, we're dealing with the multiplication of
  • the same binomial:
    • \((3x - 7)\)
  • by itself to accomplish this multiplication.

When multiplying polynomials, each term in the first polynomial is multiplied by each term in the second polynomial. This method ensures that all possible combinations of the terms are considered. For our problem:
  • Using algebraic rules, we have
    • \((3x)^2 = 9x^2\),
    • \(-2(3x)(7) = -42x\),
    • and
      • \((7)^2 = 49\).
Through careful application of rules and methods for polynomial multiplication, one can solve more complex algebraic problems efficiently.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operators. They form the building blocks of algebra. In the given problem,
  • \((3x - 7)\)
is an algebraic expression known as a binomial because it contains two terms. In algebra, the terms of an expression are separated by plus \(+\) or minus \(-\) signs.
Understanding the structure of algebraic expressions is vital for manipulating and simplifying them during calculations. Here鈥檚 a breakdown of the terms in our example:
  • **3x** is the first term, where:
    • **3** is the coefficient of the variable \(x\).
  • **-7** is the second term, a constant.
Recognizing and working with these elements is necessary to apply algebraic operations, like polynomial multiplication, effectively.
Mathematical Formulas
Mathematical formulas provide quick ways to perform complex arithmetic operations. They are essential tools in algebra for simplifying expressions and solving equations. Our task involves using the formula for squaring a binomial:
  • \((a - b)^2 = a^2 - 2ab + b^2\)
This formula helps find the square of a two-term algebraic expression efficiently.
Understanding how to apply such formulas is crucial for effective problem-solving. In our task:
  • We substitute
    • \(a = 3x\) and
    • \(b = 7\).
  • Applying the formula, we derived:
    • \((3x)^2 = 9x^2\)
    • \(-2(3x)(7) = -42x\)
    • \((7)^2 = 49\)
Mathematical formulas make solving algebraic problems faster and more reliable, enabling us to focus on more complex operations.

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Most popular questions from this chapter

Perform the indicated operations assuming all numbers are approximate. Round your answers using the procedure shown in Example 11 $$\frac{8.97(4.003)}{2.0+4.78}$$

Perform the indicated multiplications. By multiplication, show that \((x+y)^{3}\) is not equal to \(x^{3}+y^{3}\).

Make the indicated coversions. $$65.2 \frac{\mathrm{m}}{\mathrm{s}} \text { to } \frac{\mathrm{ft}}{\min }$$

Perform the indicated multiplications. In calculating the temperature variation of an industrial area, the expression \(\left(2 T^{3}+3\right)\left(T^{2}-T-3\right)\) arises. Perform the indicated multiplication.

Assume that all numbers are approximate unless stated otherwise. Find the voltage in a certain electric circuit by multiplying the sum of the resistances \(15.2 \Omega, 5.64 \Omega,\) and \(101.23 \Omega\) by the current 3.55 A.
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