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Chapter 7: Problem 85

$$ \frac{3 m^{2}}{6 m-30}-\frac{21 m-30}{6 m-30} $$

Short Answer

Expert verified
Combine the numerators over the common denominator.

Step by step solution

01

Identify Common Denominator

Notice that both fractions have the same denominator: \( 6m - 30 \). This allows us to combine the numerators over the common denominator.
02

Combine the Numerators

Combine the numerators by subtracting: \[ \frac{3m^2 - (21m - 30)}{6m - 30} \].
03

Simplify the Numerator

Distribute the negative sign across the second term of the numerator: \[ 3m^2 - 21m + 30 \].
04

Check for Factorization

Check if the numerator can be factored. Here, it does not factor nicely, so we continue with the current form.
05

Combine Numerators and Simplify

The expression now becomes: \[ \frac{3m^2 - 21m + 30}{6m - 30} \]. This is the simplified form of the given expression.

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Key Concepts

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

Common Denominator
A common denominator is a shared multiple of the denominators of two or more fractions.
When fractions have the same denominator, it becomes easy to combine them.
In the exercise, both fractions share the common denominator of \( 6m - 30 \).
To identify a common denominator, understand that it makes combining fractions possible.
  • Look at the denominators of both fractions.
  • Find the least common multiple (LCM).
Here, the common denominator \( 6m - 30 \) is already the same for both fractions.
Combining Fractions
Combining fractions involves adding or subtracting their numerators while keeping the denominator the same.
Following the common denominator, you can operate on the numerators. In our solution:
  • The numerators \(3m^2\) and \(21m - 30\) are combined.
  • Subtract \(21m - 30\) from \(3m^2\).
This gives: \( \frac{3m^2 - (21m - 30)}{6m - 30} \).
Watch for signs and distribute correctly during these steps.
Simplifying Expressions
Simplifying expressions makes them easier to work with and understand.
Simplification requires operations such as combining like terms within numerators and denominators. In this step-by-step:
  • First, handle \( 3m^2 - (21m - 30) \).
  • Distribute the negative sign across: \( 3m^2 - 21m + 30 \).
When simplified, we get the expression \(\frac{3m^2 - 21m + 30}{6m - 30}\).
Factoring Polynomials
Factoring polynomials breaks down expressions into simpler parts (factors).
These smaller parts multiply to form the original polynomial.
It鈥檚 crucial in solving and simplifying them further. In our exercise, we check if numerator: \(3m^2 - 21m + 30\) can be factored:
  • Look for common factors.
  • Apply methods like the quadratic formula (if suitable).
Unfortunately, in this case, the polynomial does not factor nicely.
Hence, the expression stays: \(\frac{3m^2 - 21m + 30}{6m - 30}\). Understanding these concepts helps in solving algebra problems effectively.

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