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Chapter 4: Problem 7

Perform each division \(\frac{12 m^{4}-6 m^{3}}{6 m^{2}}\)

Short Answer

Expert verified
The solution is \(m(2m-1)\).

Step by step solution

01

Identify the Numerator and Denominator

The given expression is \(\frac{12 m^{4}-6 m^{3}}{6 m^{2}}\). Here, the numerator is \(12 m^{4}-6 m^{3}\) and the denominator is \(6 m^{2}\).
02

Factor the Numerator

Factor out the common term in the numerator: \(12m^{4} - 6m^{3}\). The greatest common factor is \(6m^{3}\), so factor it out: \(6m^{3}(2m - 1)\).
03

Rewrite the Expression

Rewrite the expression with the factored numerator: \(\frac{6 m^{3}(2m - 1)}{6 m^{2}}\).
04

Simplify the Fraction

Simplify by canceling the common terms in the numerator and the denominator. \(\frac{6 m^{3}(2m - 1)}{6 m^{2}} = m(2m - 1)\).

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Key Concepts

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

Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler polynomials that, when multiplied together, give you the original polynomial. It is similar to finding factors of whole numbers but in the context of algebra. For example, in the expression from the exercise, the polynomial numerator is \(12m^4 - 6m^3\). This can be viewed as a combination of simpler terms. The greatest common factor (GCF) helps us to pull out the largest common term from every term in the polynomial, which is what we did when we factored out \(6m^3\) to get \(6m^3(2m - 1)\). This process makes the polynomial easier to manipulate and simplifies future calculations.
Simplifying Fractions
Simplifying fractions in algebra is like simplifying numeric fractions. The idea is to reduce the fraction to its simplest form by cancelling out common factors in the numerator and the denominator. In the exercise, once the numerator \(12m^4 - 6m^3\) was factored to \(6m^3(2m - 1)\), the expression became \(\frac{6 m^{3}(2m - 1)}{6 m^{2}}\). By dividing both the numerator and the denominator by their greatest common divisor (in this case, \(6m^2\)), we simplified the fraction to \(m(2m - 1)\). This reduced form is much simpler to work with in subsequent equations or expressions.
Common Factors in Algebra
Understanding common factors in algebra is crucial for simplifying and factoring algebraic expressions. A common factor is a term that divides two or more terms without leaving a remainder. It's the algebraic equivalent of finding the greatest common divisor. In our exercise, we identified \(6m^2\) as a common factor in both the numerator and the denominator. By cancelling out these common factors, we were able to break down the complex fraction into a simpler form. This step is essential for making further operations more manageable and less prone to errors.

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Most popular questions from this chapter

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (2 p-5)^{2} $$

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications. Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 101 \times 99 $$

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Find each product. $$ (8-3 a)(2+a) $$

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