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Chapter 3: Problem 57

Multiply. $$(x+4)(x+2)$$

Short Answer

Expert verified
Question: Multiply the two binomials \((x+4)\) and \((x+2)\) and simplify the expression. Answer: \(x^2+6x+8\)

Step by step solution

01

Apply the distributive property once

Distribute the first term in the first binomial, \(x\), to both terms in the second binomial: $$x(x+2)=x^2+2x$$
02

Apply the distributive property again

Distribute the second term in the first binomial, \(4\), to both terms in the second binomial: $$4(x+2)=4x+8$$
03

Add the two expressions

Now, add the expressions obtained in step 1 and step 2: $$(x^2+2x)+(4x+8)$$
04

Combine like terms

Combine the like terms (the \(2x\) and the \(4x\) terms): $$x^2+(2x+4x)+8$$ This results in the final expression: $$x^2+6x+8$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
Understanding the distributive property is crucial in algebra, especially when you're working on multiplying binomials like \((x+4)(x+2)\). This mathematical rule is a way to simplify complex expressions and it states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In our exercise, we apply this property twice.

First, we distribute the term \(x\) across the binomial \(x+2\), resulting in \(x^2+2x\). Next, we distribute the number \(4\) across the binomial \(x+2\), producing \(4x+8\). These steps show how the distributive property breaks down a seemingly difficult problem into manageable parts.
Combining Like Terms
Once we've distributed terms in our multiplication problem, our next step is to combine like terms. In algebra, like terms are terms that have the same variable raised to the same power. In the given expression \(x^2+2x+4x+8\), we can see that \(2x\) and \(4x\) are like terms because they both contain the variable \(x\) raised to the first power.

To combine them, we simply add their coefficients (the numbers in front of the variable) together. This results in \(2x + 4x = 6x\), significantly simplifying our expression to \(x^2+6x+8\). The process of combining like terms is essential for simplifying algebraic expressions and solving equations efficiently.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. When multiplying binomials, as in our example \((x+4)(x+2)\), we end up creating a new algebraic expression that represents the area of a rectangle with sides \(x+4\) and \(x+2\).

Through the multiplication process, using the distributive property and combining like terms, we simplify the product of the binomials to form this new expression, which in our case is \(x^2+6x+8\). Algebraic expressions are fundamental in representing real-world situations and solving problems abstractly within mathematical models.

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Most popular questions from this chapter

Multiply. $$(a-4)(a+4)$$

For Exercises, solve. A painter sections a 60 -inch by 70 -inch canvas into a grid of squares to be painted individually. What is the largest square possible so that all are of equal size and no overlapping or cutting of any region occurs?

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For Exercises, solve. The ceiling of a 45 -foot-long by 36 -foot-wide lobby of a new hotel is to be sectioned into a grid of square regions by wooden beams. What is the largest square region possible so that all regions are of equal size and no overlapping or cutting of any region takes place?

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