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Chapter 6: Problem 74

Write an equivalent expression by factoring. $$ r(t-3)-s(t-3) $$

Short Answer

Expert verified
The equivalent expression is \((t-3)(r-s)\).

Step by step solution

01

Identify Common Factor

Look at the expression and identify the common factor in both terms. Notice that \(t-3\) is present in both terms \(r(t-3)\) and \(s(t-3)\).
02

Factor Out the Common Factor

Factor the common factor \(t-3\) out of the expression. This means that \(r(t-3)-s(t-3)\) can be written as \((t-3)(r-s)\).
03

Rewrite the Factored Expression

After factoring out the common factor, the expression \(r(t-3)-s(t-3)\) is rewritten as \((t-3)(r-s)\).

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Key Concepts

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

common factor
When dealing with algebraic expressions, a common factor is a term that appears in multiple parts of the expression. Identifying common factors helps simplify and factor the expression effectively.
In the given exercise, we have the expression \(r(t-3)-s(t-3)\).

Here, we can see that \(t-3\) is a shared component in both terms.
  • The first term is \(r(t-3)\).
  • The second term is \(s(t-3)\).
The common factor \(t-3\) is crucial for simplifying the expression.
factoring
Factoring in algebra means breaking down an expression into simpler components called factors. Factors are expressions that, when multiplied together, give the original expression.
To factor an expression, follow these general steps:
  • Identify any common factors in the terms of the expression.
  • Factor out the common factor from each term.
  • Rewrite the expression using the factored form.

In our exercise, the original expression is \(r(t-3)-s(t-3)\).
We identified \(t-3\) as the common factor. By factoring out \(t-3\), the expression can be rewritten as \((t-3)(r-s)\).
equivalent expression
An equivalent expression is a different way of writing the same mathematical expression that has the same value. Factoring often leads to equivalent expressions that are simpler and easier to work with.
In our case, the given expression is \(r(t-3)-s(t-3)\).
This can be factored to produce the equivalent expression \((t-3)(r-s)\).
Although the original and factored expressions look different, they represent the same mathematical relationship.
Understanding how to convert between equivalent expressions is key in algebra for simplifying problems and solving equations efficiently.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ y^{2}-100 $$

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What is wrong with solving \(x^{2}=3 x\) by dividing both sides of the equation by \(x ?\)

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