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Chapter 5: Problem 47

Factor. $$ 4(x+y)+t(x+y) $$

Short Answer

Expert verified
The factored form is \((x+y)(4+t)\).

Step by step solution

01

Identify the Common Factor

Look at both terms of the expression. The terms are \(4(x+y)\) and \(t(x+y)\). Identify the common factor, which in this case is \((x+y)\) since it appears in both terms.
02

Factor Out the Common Factor

Now that we have identified \((x+y)\) as the common factor, we factor it out from the expression. This gives us \((x+y)(4+t)\).

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Key Concepts

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

Common Factor
Understanding the concept of a common factor is crucial when factoring algebraic expressions. A common factor in an algebraic expression is a term that appears in all parts of the expression. For example, in the expression \(4(x+y) + t(x+y)\), the part \((x+y)\) is common to both terms. By finding this common factor, you simplify the expression by factoring it out.

When factoring out a common factor, follow these steps:
  • Identify all terms in the expression.
  • Look for parts or factors common in all terms.
  • Write the common factor outside a set of parentheses.
  • Inside the parentheses, write the remaining parts of the terms when the common factor is removed.

By applying this method, the expression \(4(x+y) + t(x+y)\) simplifies to \((x+y)(4+t)\), making it easier to work with.
Distributive Property
The Distributive Property is a fundamental algebraic property that allows for the multiplication of a single term across terms within parentheses. It helps in expanding expressions and is just as vital when simplifying them, especially when working with common factors.

To apply the distributive property in reverse (which is what factoring is all about), consider the structure of the expression. For instance, recognize that \((x+y)(4+t)\) would originally have been expanded to \(4(x+y) + t(x+y)\) through distribution.

Here鈥檚 how the distributive property typically works:
  • Given an expression like \(a(b+c)\), the property states that \(a(b+c) = ab + ac\).
  • This principle allows us to rearrange terms, thus moving from a product like \((x+y)(4+t)\) back to its expanded form.

Understanding this property helps in both creating and solving equations efficiently.
Polynomials
Polynomials are expressions made up of more than one term, connected by addition or subtraction. They are a key topic in algebra, fundamental for solving equations, and are present in academic curriculums from early algebra onwards.

In the exercise \(4(x+y) + t(x+y)\), we deal with a polynomial because it's an expression consisting of multiple terms. To handle polynomials effectively:
  • Identify the individual terms separated by plus or minus signs.
  • Look for ways to factor them, typically starting by spotting common factors.
  • Apply basic algebraic properties like the distributive property to both expand and simplify them.

Factoring polynomials is a key skill for simplifying expressions, solving algebraic equations, and understanding more complex mathematical concepts.

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Most popular questions from this chapter

The polynomial function \(f(x)=0.001 x^{3}-0.12 x^{2}+3.6 x+10\) models the path of a portion of the track of a roller coaster. Use the function equation to find the height of the track for \(x=0,20,40,\) and 60 (PICUTURE NOT COPY)

Factor each expression. $$ 1-y^{6} $$

Explain why the factorization is not complete. $$ \text { Factor: } \quad 1-t^{8}=\left(1+t^{4}\right)\left(1-t^{4}\right) $$

Look Alikes... A. \(\quad 9 x^{5}-7 x^{3}+x-1\) \(+6 x^{5}-7 x^{3}+3 x+1\) \begin{tabular}{l} B. \(9 x^{5}-7 x^{3}+x-1\) \\ \(-\left(6 x^{5}-7 x^{3}+3 x-1\right)\) \\ \hline \end{tabular}

Graph the solution set of each inequality or system of inequalities on a rectangular coordinate system. $$ \left\\{\begin{array}{l} y-2<3 x \\ y+2 x<3 \end{array}\right. $$
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