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Chapter 5: Problem 30

Find each product. $$ (3 x+4 y)(3 x-4 y) $$

Short Answer

Expert verified
9x^2 - 16y^2

Step by step solution

01

- Identify the formula

Recognize that the expression (3x+4y)(3x-4y) is in the form of (a+b)(a-b) which is a difference of squares.
02

- Apply the difference of squares formula

Use the difference of squares formula (a+b)(a-b) = a^2 - b^2
03

- Substitute the values

Let a = 3x and b = 4y. Substitute to get (3x)^2 - (4y)^2
04

- Simplify the expression

Next, calculate the squares of the terms: (3x)^2 = 9x^2 and (4y)^2 = 16y^2. Thus, 9x^2 - 16y^2

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Key Concepts

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

Algebraic Multiplication
Algebraic multiplication involves multiplying two or more algebraic expressions. This is different from simple arithmetic multiplication that deals with numbers alone. When multiplying algebraic expressions, you need to consider both the coefficients and the variables.

For example, let's look at the given expression (3x+4y)(3x-4y). You are not just multiplying numbers; you are dealing with variables and coefficients. Understanding this helps in applying key formulas like the difference of squares.
Polynomial Expressions
A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. For example, both 3x and 4y are monomials (single term polynomials), while 3x + 4y is a binomial (two terms).

In the given problem, (3x+4y) and (3x-4y) are both binomials. When you multiply them, you must use the rules of polynomial multiplication. The result in this case turns out to be another polynomial (9x^2 - 16y^2).

This polynomial multiplication involved recognizing a special form called the difference of squares. Understanding these forms makes solving such problems easier.
Factoring Techniques
Factoring is a technique used to simplify polynomials by expressing them as a product of smaller polynomials. One common factoring method is recognizing the difference of squares formula: (a+b)(a-b) = a^2 - b^2.

In the given problem, you are asked to multiply (3x+4y) by (3x-4y). By identifying this as a difference of squares, you can simplify the multiplication using the formula above.

We let a = 3x and b = 4y, then: (3x+4y)(3x-4y) = (3x)^2 - (4y)^2. Calculating the squares gives: 9x^2 and 16y^2, resulting in 9x^2 - 16y^2.

Understanding this technique can help you solve similar problems quickly and efficiently.

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Most popular questions from this chapter

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 201 \times 199 $$

Find each product. \(\left(x^{2}-2\right)\left(3 x^{2}+x+4\right)\)

Multiply. $$ 4(2 a+6 b) $$

Multiply. $$ (2 c)\left(3 c^{2}\right) $$

Find each product. \((-2 k+1)\left(8 k^{2}+9 k+3\right)\)
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