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Chapter 5: Problem 63

Factor each polynomial by factoring out the GCF. $$3 x+3 y-6 z$$

Short Answer

Expert verified
The factorized form of the polynomial \(3x + 3y - 6z\) by factoring out the GCF is \(3(x+y-2z)\).

Step by step solution

01

Identify the GCF

The Greatest Common Factor (GCF) is the largest factor that divides completely into all of the terms. In this case, looking at the coefficients of the terms, the GCF is \(3\).
02

Factor out the GCF

Factorizing, 'factor out' means to 'divide out' the GCF from all terms in the polynomial. So we divide \(3x\), \(3y\), and \(-6z\) by the GCF \(3\). We then obtain \(x\), \(y\), and \(-2z\) respectively.
03

Write down the factorized form

By taking out \(3\), the expression transforms into \(3(x+y-2z)\).

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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, or GCF, is an important concept in factoring polynomials. It represents the largest number that can evenly divide all the terms of a polynomial. This is the starting point for simplifying expressions by factoring.
For example, in the polynomial expression given in the exercise: \(3x + 3y - 6z\), the coefficients are 3, 3, and -6. The greatest common factor of these numbers is 3.
  • Identify the GCF by finding the largest integer that can divide each coefficient without leaving a remainder.
  • This process simplifies the polynomial by "factoring out" this common number.
  • It ensures that the polynomial is expressed in its simplest form.
Understanding the GCF is crucial not only in this exercise but also in solving more complex polynomials where multiple steps may be needed to factor fully. Knowing how to find and work with the GCF can significantly reduce the potential for error.
Polynomials
Polynomials are algebraic expressions that consist of terms made up of variables, coefficients, and exponents. They form a cornerstone of algebra and are used extensively across different areas of mathematics.
The polynomial in this context is \(3x + 3y - 6z\), which is a linear polynomial because none of the variables have exponents greater than one.
  • A polynomial can have constants (like 3 in this case), variables \(x, y, z\), and the operations of addition, subtraction, and multiplication.
  • Each term in a polynomial has a coefficient (in this example, 3, 3, and -6 are coefficients).
  • Understanding terms and how they combine is essential for manipulating and simplifying polynomials.
Polynomials can range from simple forms like the one given to more complex expressions involving higher powers or multiple variables. Mastering the mechanics of dealing with polynomials aids greatly in algebraic problem-solving.
Algebra Concepts
Algebra encompasses a wide range of concepts and skills crucial for solving mathematical problems. Factoring polynomials, as in this exercise, is a fundamental technique in algebra.
Algebra teaches us to see relationships between numbers and variables and allows the formulation of general solutions to problems.
  • It involves operations such as addition, subtraction, multiplication, and division, extended to variables not just numbers.
  • Factoring by taking out the greatest common factor, as seen in this task, simplifies expressions and provides solutions that are easier to interpret.
  • Recognizing structure within mathematics, such as factoring and expanding, builds the foundation for understanding more advanced functions and equations.
By mastering these algebra concepts, students can tackle a variety of mathematical models and gain confidence in their analytical abilities.

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Most popular questions from this chapter

Factor. If the polynomial is prime, so indicate. $$-16 x^{4} y^{3}+30 x^{3} y^{4}+4 x^{2} y^{5}$$

Factor. $$x^{2}+6 x+9$$

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$12 x y+4 x^{2} y-72 y$$

Factor. $$a^{2}+12 a b+36 b^{2}$$

Factor. If the polynomial is prime, so indicate. $$4 a^{2}-4 a b-8 b^{2}$$
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