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Chapter 7: Problem 286

Factor completely. \(75 m^{3}+12 m\)

Short Answer

Expert verified
The complete factorization is 3m(25m^2 + 4).

Step by step solution

01

- Identify the Greatest Common Factor (GCF)

Determine the greatest common factor of the terms in the expression. For the expression 75m^3 + 12m, both terms have a common factor of 3m.
02

- Factor Out the GCF

Divide each term by the GCF (3m) and rewrite the expression: 75m^3 梅 3m = 25m^2 12m 梅 3m = 4 So, 75m^3 + 12m can be factored as 3m(25m^2 + 4).
03

- Verify the Factorization

Distribute 3m back into the factored expression to check: 3m(25m^2 + 4) = 75m^3 + 12m. The expression matches the original, verifying the factorization is correct.

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) is the largest number that divides two or more terms without leaving a remainder. Finding the GCF is crucial in simplifying expressions and factoring polynomials.

In our example, the terms are 75m^3 and 12m. To find the GCF:
  • Identify the common numerical factor: both 75 and 12 are divisible by 3.
  • Identify the common variable factor: both terms contain the variable 'm'.
    So, the GCF of 75m^3 and 12m is 3m.
The GCF helps in reducing the polynomial to its simplest form by removing the common factors.
Factoring Technique
Factoring involves breaking down a complex expression into simpler multipliers or factors. This technique is widely used to simplify algebraic expressions and solve equations more easily. Here鈥檚 how it works for our example:

After identifying the GCF (3m), we factor it out from the polynomial. This means dividing each term by the GCF:
  • 75m^3 梅 3m = 25m^2
  • 12m 梅 3m = 4
Therefore, the factored form of 75m^3 + 12m is 3m(25m^2 + 4).

By factoring out the GCF, the expression becomes simpler and more manageable.
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It allows us to express and solve problems involving unknown values and complex relationships. In the context of our example:

We use algebraic techniques to factor and simplify the polynomial. By working through the problem, we:
  • Identify the GCF
  • Factor the polynomial by dividing it by the GCF
  • Verify our work by distributing the GCF back into the factored expression and ensuring it matches the original
These steps illustrate fundamental algebraic principles that help solve a wide range of mathematical problems, making algebra a powerful and essential tool for students.

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