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Chapter 6: Problem 30

Divide and check. $$ \frac{3 x^{2}-7 x}{3 x-1} $$

Short Answer

Expert verified
\(\frac{x(3x - 7)}{3x - 1}\)

Step by step solution

01

Factor the numerator

First, factor the numerator if possible. The numerator is given as \(3x^2 - 7x\). Factor out the common term, which is \(x\):\[3x^2 - 7x = x(3x - 7)\]
02

Rewrite the expression

Substitute the factored form of the numerator back into the original division problem:\[\frac{3x^2 - 7x}{3x - 1} = \frac{x(3x - 7)}{3x - 1}\]
03

Simplify the fraction

Since \(3x - 7\) does not cancel with \(3x - 1\), the fraction cannot be further simplified. The simplified form remains:\[\frac{x(3x - 7)}{3x - 1}\]
04

Check the division

Multiply the denominator by the simplified quotient to check if it equals the numerator:\(3x - 1 \cdot \frac{x(3x - 7)}{3x - 1} = x(3x - 7)\). Since this matches the factored numerator, the simplification is correct.

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is a useful technique in algebra that simplifies expressions and helps in solving equations.
When factoring, we look for common factors in the terms of the polynomial and express the polynomial as a product of these factors.
Let's take the expression we have: \(3x^2 - 7x\).
Both terms share a common factor of \(x\). By factoring out \(x\), we write:\[3x^2 - 7x = x(3x - 7)\]
Factoring transforms more complex expressions into simpler, more manageable pieces, making it easier to perform further operations like division.
Simplifying Rational Expressions
Simplifying rational expressions involves reducing fractions to their simplest form.
A rational expression is a fraction where both the numerator and the denominator are polynomials, such as \( \frac{3x^2 - 7x}{3x - 1} \).
After factoring the numerator as \( x(3x - 7) \), we rewrite the expression as:
\( \frac{x(3x - 7)}{3x - 1} \).
To simplify, we check if any terms in the numerator and the denominator can cancel each other out. Here, the term \(3x - 7\) doesn't cancel out with \(3x - 1\), thus the fraction remains simplified as it is.
Simplification makes expressions easier to work with and is essential for accurately solving more complex algebraic equations.
Checking Solutions
Once we've simplified an expression, it's important to check our work to ensure accuracy.
Here, we want to confirm that our simplified expression \( \frac{x(3x - 7)}{3x - 1} \) is correct.
We do this by multiplying the denominator \(3x - 1\) by the simplified expression and see if we get back the original numerator:
\[ (3x - 1) \cdot \frac{x(3x - 7)}{3x - 1} = x(3x - 7) \].
By performing the multiplication and verifying that the product equals the original numerator, we ensure our division and simplification are correct.
Checking solutions is a crucial step in problem-solving as it confirms the validity of our results and helps avoid errors.

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Most popular questions from this chapter

Knitted fabric is described in terms of wales per inch (for the fabric width) and courses per inch (CPI) (for the fabric length). The CPI \(c\) is inversely proportional to the stitch length \(l .\) For a specific fabric with a stitch length of 0.166 in. the CPI is \(34.85 .\) What would the CPI be if the stitch length were increased to 0.175 in.?

Find an equation of variation in which: \(y\) varies directly as the square of \(x,\) and \(y=50\) when \(x=10\)

Simplify. Do not use negative exponents in the answer. $$ \left(-5 x^{2} y^{-6}\right)^{-3} $$

If \(y\) varies inversely as the cube of \(x\) and \(x\) is multiplied by \(0.5,\) what is the effect on \(y ?\)

Find the variation constant and an equation of variation in which \(y\) varies inversely as \(x,\) and the following conditions exist. \(y=9\) when \(x=10\)
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