Problem 29 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(3 x+4)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(9x^2 + 24x + 16\).

Step 1: Identify the Special Product Formula

The expression \((3x+4)^2\) is in the form of \((a+b)^2\), which is a special product formula known as the square of a binomial. This formula is \((a+b)^2 = a^2 + 2ab + b^2\).

Step 2: Assign values to a and b

In the expression \((3x+4)^2\), assign \(a = 3x\) and \(b = 4\). These values will be used in the special product formula to expand the binomial.

Step 3: Apply the Special Product Formula

Substitute \(a = 3x\) and \(b = 4\) into the formula \((a+b)^2 = a^2 + 2ab + b^2\). This gives:\((3x+4)^2 = (3x)^2 + 2(3x)(4) + 4^2\).

Step 4: Simplify Each Term

Calculate each term separately:- \((3x)^2 = 9x^2\)- \(2(3x)(4) = 24x\)- \(4^2 = 16\)

Step 5: Combine All Terms

Combine the simplified terms to get the final expression: \(9x^2 + 24x + 16\)

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a major part of algebra, where numbers and variables are combined using various operations such as addition, subtraction, multiplication, and division. An algebraic expression represents a mathematical phrase that can include constants, coefficients, variables, and operators. For instance, in the expression \(3x + 4\), there are three components to note:

• Constants: Fixed values, like 4 in the expression.
• Variables: Symbols that represent an unknown value, such as \(x\).
• Coefficients: Numbers that are multiplied by the variables, like 3 in \(3x\).

Algebraic expressions can be manipulated using different algebraic operations. This allows us to simplify or expand expressions, solve equations, and ultimately understand more complex mathematical relationships. Recognizing these parts helps in simplifying and manipulating the expressions in various algebraic problems.

Exploring Binomials

A binomial is a specific type of algebraic expression that consists of only two terms. These terms are usually linked together by a plus or minus sign. An example of a binomial expression is \(3x + 4\), where there are two distinct components:

• The first term \(3x\).
• The second term 4.

Binomials can exhibit intriguing properties, especially when used in polynomial operations. One of the most useful properties of binomials is found in the "Special Product Formulas," which allow us to expand and simplify expressions more efficiently.

When dealing with the square of a binomial like \((3x+4)^2\), a special product formula helps to compute the expression without performing complex multiplications directly. Understanding how to identify and apply these formulas is crucial for simplifying such expressions.

Basics of Polynomial Expansion

Polynomial expansion involves expressing a polynomial in a more extended form by using various algebraic techniques. The concept is often applied using special product formulas that simplify the process.

In the prompt exercise, the square of a binomial \((3x+4)^2\) is expanded using the formula \((a+b)^2 = a^2 + 2ab + b^2\). This specific formula allows us to break down the binomial expansion without direct multiplication.Let's see a breakdown of this process:

• Identify each component of the binomial: \(a = 3x\) and \(b = 4\).
• Calculate each component squared: \((3x)^2 = 9x^2\) and \(4^2 = 16\).
• Calculate the middle term: \(2 \times 3x \times 4 = 24x\).
• Combine all parts to get the expanded polynomial: \(9x^2 + 24x + 16\).

This formula is beneficial because it offers a systematic way to simplify expressions and understands the behavior of polynomials in algebra.