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Chapter 4: Problem 29

Find each product. $$ (5 y+3 x)(5 y-3 x) $$

Short Answer

Expert verified
(5y + 3x)(5y - 3x) = 25y^2 - 9x^2

Step by step solution

01

Recognize the Pattern

Notice that the given expression (5y + 3x)(5y - 3x) is of the form (a + b)(a - b) , where a = 5y and b = 3x.
02

Apply the Difference of Squares Formula

The difference of squares formula is given by (a + b)(a - b) = a^2 - b^2. Apply this formula using a = 5y and b = 3x. So, (5y + 3x)(5y - 3x) = (5y)^2 - (3x)^2.
03

Calculate the Squares

Square the terms individually: (5y)^2 = 25y^2 and (3x)^2 = 9x^2.
04

Combine the Results

Now substitute the squared terms back into the difference of squares formula: (5y + 3x)(5y - 3x) = 25y^2 - 9x^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 is a basic concept in algebra. It involves multiplying variables and constants together. When multiplying terms, it's important to remember to apply distributive property correctly. For example, if you have an expression like \(a \times b\), you multiply the numerical coefficients and variables separately.
In algebraic expressions, it's crucial to simplify your results. For instance, \(5y \times 3x = 15yx\).
Multiplying terms might sometimes involve special formulas like the difference of squares, which we will discuss later in the article.
Polynomial Expressions
Polynomial expressions are mathematical expressions that include variables raised to whole number exponents and coefficients. A polynomial can have terms like \(ax^n\), where \(a\) is a coefficient, \(x\) is the variable, and \(n\) is a non-negative integer exponent.
For example, \(5y + 3x\) or \(2x^2 - 4x + 7\) are polynomial expressions. They can be added, subtracted, and multiplied to form new polynomials.
Knowing how to properly combine like terms and distribute coefficients helps simplify these expressions, making them easier to work with in algebra.
Product of Binomials
The product of binomials is when you multiply two binomial expressions together. Binomials are expressions with two terms, like \(a + b\) and \(a - b\). A common multiplication method for binomials is the FOIL method, which stands for First, Outer, Inner, Last.
For example, to multiply \( (a + b)(a - b) \):
  • First: \(a \times a = a^2 \)
  • Outer: \(a \times -b = -ab \)
  • Inner: \(b \times a = ba \)
  • Last: \(b \times -b = -b^2 \)
Simplify to get \(a^2 - b^2\).
In the given exercise, we notice it follows this pattern with \(a = 5y \) and \(b = 3x \).
Squares of Variables
A square of a variable simply means multiplying the variable by itself. This is often denoted as \(x^2\). For example, squaring \(5y\) gives \( (5y)^2 = 25y^2 \). Similarly, squaring \(3x\) results in \( (3x)^2 = 9x^2 \).
These squared terms maintain the variable's identity while elevating its power. They play a crucial role in polynomial expressions and algebraic multiplication.
In our particular problem, we apply this to get the final simplified result: \((5y)^2 - (3x)^2 = 25y^2 - 9x^2\).

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Most popular questions from this chapter

Find each product. $$ -3 k\left(8 k^{2}-12 k+2\right) $$

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(m^{6} n\right)^{-2}\left(m^{2} n^{-2}\right)^{3}}{m^{-1} n^{-2}} $$

Our system of numeration is called a decimal system. In a whole number such as 2846 each digit is understood to represent the number of powers of 10 for its place value. The 2 represents two thousands \(\left(2 \times 10^{3}\right),\) the 8 represents eight hundreds \(\left(8 \times 10^{2}\right),\) the 4 represents four tens \(\left(4 \times 10^{1}\right),\) and the 6 represents six ones (or units) \(\left(6 \times 10^{\circ}\right)\) \(2846=\left(2 \times 10^{3}\right)+\left(8 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(6 \times 10^{0}\right) \quad\) Expanded form $$ \text {Keeping this information in mind,} $$ Divide 2846 by \(2,\) using paper-and-pencil methods: \(2 \longdiv { 2 8 4 6 }\)

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ (5 k+3 q)^{2} $$

Subtract. See Example \(\boldsymbol{\delta}\) $$ \begin{array}{l} {12 m^{3}-8 m^{2}+6 m+7} \\ {-3 m^{3}+5 m^{2}-2 m-4} \\ \hline \end{array} $$
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