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Chapter 5: Problem 26

\((7 x+5 y)(x+y)\)

Short Answer

Expert verified
The simplified form of the algebraic expression \((7x + 5y)(x+y)\) is \(7x^2 + 12xy + 5y^2\)

Step by step solution

01

Distribute the First Term in the First Parenthesis

Apply the distributive law to the first term in the first parenthesis, \(7x\), resulting in \(7x*x + 7x*y\). This simplifies to \(7x^2 + 7xy\)
02

Distribute the Second Term in the First Parenthesis

Apply the distributive law to the second term in the first parenthesis, \(5y\), resulting in \(5y*x + 5y*y\). This simplifies to \(5yx + 5y^2\). However, since multiplication is commutative, the term \(5yx\) could be also written as \(5xy\)
03

Combine Like Terms

Combine like terms from Step 1 and Step 2 resulting in the final expression \(7x^2 + 7xy + 5xy + 5y^2\). This simplifies further to \(7x^2 + 12xy + 5y^2\)

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Key Concepts

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

Polynomial Multiplication
Polynomial multiplication involves expanding expressions by applying the distributive property. In the expression \((7x+5y)(x+y)\), we first multiply each term inside the first parenthesis by every term in the second parenthesis.
  • Distributive Property: This is a mathematical rule used to multiply a single term across multiple terms within parentheses. It allows us to simplify expressions efficiently.
  • Example: When distributing, \(7x\) multiplies each term in \((x+y)\), giving us \(7x \cdot x + 7x \cdot y\), which simplifies to \(7x^2 + 7xy\).
  • Repeat with the Next Term: We then do the same with \(5y\), resulting in \(5y \cdot x + 5y \cdot y\), or \(5xy + 5y^2\).
Breaking the multiplication into steps helps avoid mistakes and ensures all terms are accounted for before simplifying.
Like Terms
In algebra, like terms are terms that have the same variables raised to the same exponents. Identifying like terms is crucial to simplifying expressions by combining them.
  • Definition: Like terms must match in both their variable part and exponent. For example, \(7xy\) and \(5xy\) are like terms, while \(7xy\) and \(5x^2y\) are not.
  • Combining Like Terms: Adding or subtracting like terms involves summing their coefficients. In the final step of the exercise, we combine \(7xy + 5xy\) to get \(12xy\).
Recognizing and combining like terms simplifies algebraic expressions and is a fundamental skill in algebra.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Unlike equations, expressions don't have an equal sign.
  • Components: Variables, coefficients, and terms make up algebraic expressions. Variables represent unknown values, coefficients are numbers multiplying the variables, and terms are parts of the expression separated by addition or subtraction.
  • Simplification: The ultimate goal is to simplify these expressions into a form with fewer terms, whenever possible. By performing operations and combining like terms, we make expressions easier to work with.
  • Example: The simplified expression from our exercise, \(7x^2 + 12xy + 5y^2\), is an algebraic expression composed of three separate terms.
Mastering algebraic expressions empowers students to solve more complex problems in geometry, calculus, and beyond.

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Most popular questions from this chapter

\((9 x-5)-(x+7)\)

\(\frac{12 l^{5}-6 l^{2}+2 l}{2 l}\)

Alternate Methods Describe a method that you can use to multiply any two polynomials.

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