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Magazine Chapter 4: Problem 38
In Exercises 35鈥�42, fi nd the product. \((2 y-5)(2 y+5)\)
Short Answer
Expert verified
The product of \((2y-5)(2y+5)\) is \(4y^2 - 25\).
Step by step solution
01
Distribute the First Terms
First, multiply the first terms of each binomial together: \(2y * 2y = 4y^2\).
02
Distribute the Outer Terms
Then, multiply the outer terms of each binomial together: \(2y * 5 = 10y\).
03
Distribute the Inner Terms
Next, multiply the inner terms of each binomial together: \(-5 * 2y = -10y\).
04
Distribute the Last Terms
Lastly, multiply the last terms of each binomial together: \(-5 * 5 = -25\).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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: \(ac+ad+bc+bd\).
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:
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 \(-5 * 5\).
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.
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.
- Lorem ipsum dolor sit amet, consectetur adipiscing elit.
- 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 \(4y^2 - 25\).
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.
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.
