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Chapter 6: Problem 236

Factor completely. $$ 75 m^{3}+12 m $$

Short Answer

Expert verified
3m(25m^{2}+4)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, find the greatest common factor (GCF) of the terms in the expression. The given terms are 75m^{3} and 12m. The numerical GCF of 75 and 12 is 3, and each term contains at least one m . Therefore, the GCF is 3m.
02

Factor out the GCF

75m^{3}+12m = 3m(25m^{2}+4)

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Key Concepts

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

greatest common factor
When working with polynomial expressions, one of the first steps in factoring is to identify the greatest common factor (GCF). The GCF is the largest factor that divides all the terms in the polynomial. This includes both numbers and any common variables.

For the expression given in the exercise: 75m^{3} + 12m, we need to break down each term to determine their common factors.
  • First, examine the coefficients: 75 can be factored into 3 脳 25, and 12 can be factored into 3 脳 4. The common factor between 75 and 12 is 3.
  • Next, look at the variable part: m^{3} and m both have at least one m. Therefore, the common variable factor is m.

Combining these results, the GCF is 3m. Identifying the GCF simplifies the polynomial, making the next steps in factoring easier.
polynomial expressions
Polynomials are algebraic expressions that consist of terms in the form of ax^{n}, where 'a' is a coefficient, and 'n' is a non-negative integer exponent. Each term in a polynomial can have both a numerical coefficient and variables raised to positive integer powers.

In the exercise, we are given the polynomial expression: 75m^{3} + 12m. This expression has two terms:
  • 75m^{3}
  • 12m

Each term is composed of a coefficient (75 and 12 respectively) and the variable 'm' raised to different powers (m^{3} and m^{1}). Understanding how to manipulate these expressions is crucial in algebra, especially when it comes to simplifying and solving equations.
algebraic factoring
Algebraic factoring allows you to simplify polynomial expressions by expressing them as the product of simpler polynomials or factors. Factoring is an essential skill in algebra as it can reveal crucial information about the roots or solutions of the polynomial.

For the provided expression, 75m^{3} + 12m, the factoring process involves these steps:
  • First, we identified the GCF as 3m.
  • Next, we factor out the GCF from each term: 75m^{3} + 12m = 3m(25m^{2} + 4).

Factoring out the GCF simplifies the polynomial and often makes further operations, like solving equations, more manageable.

By practicing these processes, you'll be able to factor more complex polynomial expressions efficiently.

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Most popular questions from this chapter

In the following exercises, factor by grouping. $$ r^{2}-3 r-r+3 $$

In the following exercises, factor each expression using any method. $$ 12 y^{2}-29 y+14 $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ m^{2}-13 m+30 $$

Tommy factored \(x^{2}-x-20\) as \((x+5)(x-4)\). Sara factored it as \((x+4)(x-5)\). Ernesto factored it as \((x-5)(x-4)\). Who is correct? Explain why the other two are wrong.

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ q^{2}-13 q+36 $$
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