Problem 73 - College Algebra Essentials

Question

In Exercises 67–82, find each product. $$(7 x+5 y)^{2}$$

Step-by-Step Solution

Answer

The product \( (7 x+5 y)^{2} \) equals \( 49x^2 + 70xy + 25y^2 \).

Step 1: Identify a and b in the binomial

In the provided binomial \((7 x+5 y)^{2}\), \(a\) is \(7x\) which represents the first term and \(b\) is \(5y\) which represents the second term. Now apply the formula for the square of a binomial which is \((a+b)^{2}=a^{2}+2ab+b^{2}\).

Step 2: Apply the square of a binomial formula

Squaring the first and second terms, then multiplying both terms by 2, gives: \(a^{2} = (7x)^2\), \(2ab = 2*7x*5y\) and \(b^{2} = (5y)^2\). This gives us a new expression \(49x^2 + 70xy + 25y^2\).

Step 3: Combine all terms

Then the result after substituting all expressions back into the formula is the sum, which is the final answer. Write all the terms together, so the final expression is \(49x^2 + 70xy + 25y^2\).

Key Concepts

Square of a Binomial

Squaring a binomial involves expanding an expression that appears in the form \((a+b)^2\). This is a common concept in algebra. When you square a binomial, you use a specific formula to find the product:

• \((a+b)^2 = a^2 + 2ab + b^2\)

This formula indicates that you square the first term, double the product of both terms, and then square the second term.
For instance, with the binomial \((7x + 5y)^2\), \(a\) is \(7x\), and \(b\) is \(5y\). Squaring each term gives us \(49x^2\) and \(25y^2\).
Doubling the product of \(a\) and \(b\) results in \(70xy\), giving the final expanded expression as \(49x^2 + 70xy + 25y^2\).
Understanding this formula helps simplify what may initially seem like a complex expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. They are fundamental in representing real-world scenarios where relationships between varying quantities need to be described.
In an algebraic expression, variables stand in for unknown or changeable values, and numbers can be constants or coefficients that scale the variables.
For example, in the expression \(7x + 5y\), \(7\) and \(5\) are coefficients that tell us how much of each variable, \(x\) and \(y\), is present. The combination of variables and coefficients through addition, subtraction, multiplication, and sometimes division or exponentiation, allows us to express a wide range of mathematical problems and solutions.
These expressions can be expanded, factored, solved, or simplified depending on what is being asked in a problem. Mastery in handling algebraic expressions is essential for tackling more complex mathematical operations.

Multiplication of Polynomials

Multiplying polynomials involves distributing each term in one polynomial to every term in the second polynomial. This is an intricate skill and requires practice to perform efficiently and correctly.
For the sake of simplicity, let's look at a simple case—the multiplication of binomials. If we consider two binomials \((a+b)\) and \((c+d)\), expanding them requires multiplying each term from the first binomial by each term of the second:

• \(ac\)
• \(ad\)
• \(bc\)
• \(bd\)

The outcome is a new polynomial expression, \(ac + ad + bc + bd\).
Multiplying polynomials is about ensuring every term is accounted for and accurately combined.
In some cases, like squaring a binomial, the process is simplified using formulas like \((a+b)^2 = a^2 + 2ab + b^2\). This avoids random errors and helps maintain a structured approach to finding the correct expression.