Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blanks to describe each special product. a. \((x+y)^{2}=x^{2}+2 x y+y^{2}\) The _____ of the second term _____ the product of the first and second terms. The square of the _____ term. b. \((x+y)(x-y)=x^{2}-y^{2}\) The square of the _____ term. The _____ of the first term.

Step-by-Step Solution

Answer

a. The square, plus; first. b. first; negative.

Step 1: Understanding Part a

In the expression (x+y)^{2}=x^{2}+2xy+y^{2}, the terms are as follows: - The first term is x^{2}, which is the square of the first term. - The second term is 2xy, which is double the product of the first and second terms. - The third term is y^{2}, which is the square of the second term.

Step 2: Fill in Blanks for Part a

Using the understanding from Step 1, fill in the blanks: - The **square** of the second term **plus** the product of the first and second terms. - The square of the **first** term.

Step 3: Understanding Part b

For the expression (x+y)(x-y)=x^{2}-y^{2}: - The first term is x^{2}, which is the square of the first term. - The second term -y^{2} is the negative of the square of the second term.

Step 4: Fill in Blanks for Part b

From Step 3, fill in the blanks: - The square of the **first** term. - The **negative** of the first term.

Key Concepts

Square of Binomials

In algebra, the **square of binomials** is a special product. A binomial is simply a mathematical expression with two terms, such as \((x + y)\). When you square a binomial, you multiply the binomial by itself:

• \((x + y)^2\) becomes \((x + y)(x + y)\).

To calculate this, you apply the distributive property:

• First, multiply the first term of the first binomial by each term of the second binomial.
• Next, multiply the second term of the first binomial by each term of the second binomial.
• Combine like terms.

Doing so gives you the famous formula for the square of a binomial:

• \((x + y)^2 = x^2 + 2xy + y^2\).

This results from:- \(x^2\), which is the square of the first term.- \(2xy\), which is twice the product of the first and second terms.- \(y^2\), which is the square of the second term.
Once you understand this pattern, working with binomial squares becomes much easier!

Difference of Squares

Another key special product is the **difference of squares**. Here, you deal with expressions like \((x + y)(x - y)\).

• This results in \(x^2 - y^2\).

This occurs because:- The multiplication involves canceling out the middle terms.- You multiply \(x\) by \(x\) to get \(x^2\), and \(-y\) by \(y\) to get \(-y^2\).

• Other terms like \(xy\) and \(-xy\) cancel each other out, leaving you with just \(x^2 - y^2\).

The formula for the difference of squares is:

• \((x + y)(x - y) = x^2 - y^2\).

This happens due to the symmetric structure of the binomials and the nature of the operation. Knowing this formula can save you a lot of time when solving algebraic expressions!

Algebraic Expressions

At its core, an **algebraic expression** is simply a combination of numbers, variables, and mathematical operations. The expressions can include:

• Variables like \(x\), \(y\), or \(z\).
• Coefficients, which are constants multiplying the variables.
• Operations like addition, subtraction, multiplication, and division.

When dealing with algebraic expressions, it's crucial to follow the order of operations, which helps in simplifying expressions correctly.

• Special products like square of binomials and difference of squares are examples of how algebraic expressions can be simplified using formulas.

Understanding algebraic expressions involves recognizing patterns and using known formulas to make the simplification process quick and efficient. This is particularly useful when working with polynomials or solving equations in algebra.