Question

In Exercises 35–42, fi nd the product. \((2 y-5)(2 y+5)\)

Short answer

The product of \((2y-5)(2y+5)\) is \(4y^2 - 25\).

Step-by-step solution

Distribute the First Terms

First, multiply the first terms of each binomial together: \(2y * 2y = 4y^2\).

Distribute the Outer Terms

Then, multiply the outer terms of each binomial together: \(2y * 5 = 10y\).

Distribute the Inner Terms

Next, multiply the inner terms of each binomial together: \(-5 * 2y = -10y\).

Distribute the Last Terms

Lastly, multiply the last terms of each binomial together: \(-5 * 5 = -25\).

Combine Like Terms

Combine the like terms: \(4y^2 + 10y - 10y - 25\), this simplifies to \(4y^2 - 25\) because the \(+10y\) and \(-10y\) cancel out.

Key concepts

Binomial Multiplication

Binomial multiplication is an important process in algebra. A binomial is a polynomial with two terms. Multiplying two binomials may seem challenging, but it follows a simple pattern. The goal is to use the distributive property on each term in the first binomial to the terms in the second.

Here's how it works:

• Write down both binomials, for example, \((a+b)(c+d)\).
• Multiply each term from the first binomial with every term in the second binomial.
• In our example, multiply as follows: first with first, outer with outer, inner with inner, and last with last.
• The result is: \(ac+ad+bc+bd\).

This method is often remembered by the acronym FOIL, which stands for First, Outer, Inner, Last. It's a helpful way to remember the order of operations when multiplying binomials.

Distributive Property

The distributive property is a fundamental rule used in algebra. It allows you to multiply a single term and two or more terms inside a parenthesis. This method ensures all terms are accounted for when multiplying.

In the expression \((2y-5)(2y+5)\), the distributive property means multiplying each term in the first binomial by each term in the second binomial:

• Start with the first term from the first binomial, \(2y\), and distribute it over both terms in the second binomial: \(2y * 2y\) and \(2y * 5\).
• Then take the second term from the first binomial, \(-5\), and distribute it over the same terms: \(-5 * 2y\) and \(-5 * 5\).

Following these steps ensures every combination of terms gets multiplied. The result before combining like terms would be \(4y^2 + 10y - 10y - 25\). Using the distributive property simplifies complex multiplication into manageable steps.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. Terms in a polynomial are considered "like" if they have the same variables raised to the same powers.

In our exercise, after distributing, we got \(4y^2 + 10y - 10y - 25\). Here, \(10y\) and \(-10y\) are like terms because they both contain \(y\) with the exponent of 1.

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• Different terms with the same variable and exponent should be combined by adding or subtracting their coefficients.
• In our case, \(10y - 10y\) cancels out, leaving just \(4y^2 - 25\).

By combining like terms, the expression becomes simpler and more manageable, leading to the final result.

Polynomials

Polynomials are expressions that consist of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents on variables.

A basic polynomial might look like \(ax^n + bx^{n-1} + \, ... \, + c\). When dealing with binomials like \((2y-5)\) and \((2y+5)\), we use polynomial multiplication to find a more complex polynomial.

• Polynomials can be classified based on their number of terms: monomials (1), binomials (2), trinomials (3), etc.
• The degree of a polynomial is determined by the highest exponent of its variable.
• Our exercise simplifies from a product of binomials to a single polynomial, \(4y^2 - 25\), where the highest degree is 2.

Battling through the processes of multiplication, distribution, and combining terms, we're left with a neat polynomial, giving a comprehensive summary of the algebraic expression.