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Chapter 6: Problem 37

Expand each expression. Simplify your results by combining like terms. \(2(x+y)+x(3+y)(x+2)\)

Short Answer

Expert verified
3x^2 + x^2 y + 8x + 2y + 2xy

Step by step solution

01

- Distribute the constants and variables in first term

Expand the expression by distributing the 2 into the terms inside the parenthesis: 2(x+y) = 2x + 2y
02

- Expand the second term

Expand the second term by applying the distributive property step-by-step. First, expand (3+y)(x+2): 3(x+2) + y(x+2) = 3x + 6 + yx + 2y
03

- Multiply the expanded second term by the outer x

Now multiply the expanded terms by x: x(3x + 6 + yx + 2y) = 3x^2 + 6x + x^2 y + 2xy
04

- Combine all expanded terms from both expressions

Combine the results from Step 1 and Step 3: 2x + 2y + 3x^2 + 6x + x^2 y + 2xy
05

- Simplify by combining like terms

Identify and combine like terms: 3x^2 + x^2 y + 8x + 2y + 2xy

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Key Concepts

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

Distributive Property in Algebra
The distributive property is a basic yet powerful algebraic principle. It states that to multiply a sum by a number, you can multiply each addend by the number and then add the products. In mathematical terms, for any numbers or variables a, b, and c, the distributive property can be written as: a(b + c) = ab + ac. In our exercise, we used the distributive property to expand the expression 2(x + y) into 2x + 2y. This means we multiplied both x and y by 2. We also used this property for the term x(3 + y)(x + 2). We first expanded (3 + y)(x + 2) by multiplying each term separately. This resulted in 3(x + 2) + y(x + 2), which we then simplified further.
Combining Like Terms
Combining like terms is an essential step when simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. Only these like terms can be combined by adding or subtracting their coefficients. In our example, after expanding all parts of the expression, we ended up with: 2x + 2y + 3x^2 + 6x + x^2y + 2xy. Here, we need to combine the terms that have the same variables and powers. By identifying the like terms, 2x and 6x are combined to give 8x. The terms that do not have any like terms, such as 3x^2, x^2y, 2y, and 2xy, remain as they are in the simplified expression: 3x^2 + x^2y + 8x + 2y + 2xy.
Polynomial Expansion
Expanding polynomials involves applying the distributive property repeatedly until all terms are multiplied out. In our case, the original expression had multiple components that needed to be expanded separately: 2(x + y) + x(3 + y)(x + 2). First, we expanded 2(x + y) using the distributive property. Then, we focused on expanding x(3 + y)(x + 2) in parts. We broke it down to first expand (3 + y)(x + 2), and then multiply the resulting expression by x. This careful step-by-step expansion ensures that all terms are accounted for and correctly combined. Polynomial expansion is essential for simplifying complex algebraic expressions and making them manageable.

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