Question

Use the Distributive Property to remove the parentheses. $$ (x+5)(x+1) $$

Short answer

The simplified expression is \[x^2 + 6x + 5 \]

Step-by-step solution

Identify the Expression

Recognize the two binomials being multiplied: \(x+5\) and \(x+1\).

Apply the Distributive Property for the First Term

Multiply \(x\) from the first binomial by each term in the second binomial: \[ x \times (x + 1) = x^2 + x \]

Apply the Distributive Property for the Second Term

Multiply \(5\) from the first binomial by each term in the second binomial: \[ 5 \times (x + 1) = 5x + 5 \]

Combine Like Terms

Add the results from the two distributive steps: \[ x^2 + x + 5x + 5 \]

Simplify the Expression

Combine the like terms \(x\) and \(5x\) to get the final simplified expression: \[ x^2 + 6x + 5 \]

Key concepts

Binomials

Let's start with understanding what binomials are. Binomials are algebraic expressions that contain two terms. For example, in the exercise, we have two binomials: \(x+5\) and \(x+1\). These are called binomials because they have two terms each. To identify binomials, look for expressions that are sums or differences of two distinct terms. Make sure to recognize these structures because they frequently appear in polynomial multiplication and other algebraic operations.

Combining Like Terms

Combining like terms is essential for simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. In the exercise, after we used the distributive property, we got the expression: \[x^2 + x + 5x + 5\]. To simplify, we combine the like terms \(x\) and \(5x\):
\[x + 5x = 6x\].
This gives us the final simplified expression: \[x^2 + 6x + 5\].
Remember, combining like terms makes expressions simpler and easier to work with, especially in algebraic equations and polynomial functions.

Multiplying Polynomials

Multiplying polynomials involves applying the distributive property. This property allows you to multiply each term in one polynomial by each term in the other polynomial. In our exercise, we multiplied \(x+5\) by \(x+1\).
First, multiply \(x\) by each term in \(x+1\):
\[x \times (x + 1) = x^2 + x\].
Next, multiply \(5\) by each term in \(x+1\):
\[5 \times (x + 1) = 5x + 5\].
After distributing, we combine all the terms: \[x^2 + x + 5x + 5\].
Lastly, we combine like terms to simplify the expression to \[x^2 + 6x + 5\].
Mastering polynomial multiplication is foundational for more advanced algebra topics and helps in solving higher-degree equations.