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Chapter 5: Problem 21

Factor out the GCF in each polynomial. $$ 3 x\left(6 x^{2}+5\right)-2\left(6 x^{2}+5\right) $$

Short Answer

Expert verified
The factored expression is \((6x^2 + 5)(3x - 2)\).

Step by step solution

01

Identify Common Factors

First, observe the terms in the polynomial: \(3x(6x^2 + 5)\) and \(-2(6x^2 + 5)\). Notice that both terms contain the common factor \((6x^2 + 5)\). This is the greatest common factor (GCF) present in both terms.
02

Factor Out the GCF

Since \((6x^2 + 5)\) is the common factor, factor it out of the expression: \( (6x^2 + 5) \left(3x - 2\right) \). Here, dividing each term by \((6x^2 + 5)\) results in the other factor, \(3x - 2\).
03

Verify the Factored Form

Multiply the factored expression back together to ensure it equals the original polynomial. Expand \((6x^2 + 5)(3x - 2)\):1. \((6x^2 + 5) \times 3x = 18x^3 + 15x\)2. \((6x^2 + 5) \times -2 = -12x^2 -10\)3. Combined, this gives you the original expression: \(3x(6x^2 + 5) - 2(6x^2 + 5)\).

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Key Concepts

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

Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is an essential concept in mathematics that helps simplify expressions and solve equations more easily. When dealing with polynomials, identifying the GCF involves searching for the largest factor that is common to all terms present. Here's how you can find it for polynomials:
  • First, analyze each term of the polynomial individually. For instance, take the polynomial expression from the exercise: \(3x(6x^2 + 5) - 2(6x^2 + 5)\).
  • Both terms鈥擻(3x(6x^2 + 5)\) and \(-2(6x^2 + 5)\)鈥攕hare a common sub-expression: \((6x^2 + 5)\). This sub-expression is the GCF for the given polynomial.
Finding the GCF simplifies the factoring process significantly. By knowing the GCF, you can reduce the complexity of the polynomial, making calculations easier and more efficient.
Polynomial Expressions
Polynomial expressions consist of variables raised to various powers, combined using addition, subtraction, and multiplication. They can have one or multiple terms, called monomials, each made up of coefficients and variables.
  • For example, consider the polynomial expression \(3x(6x^2 + 5) - 2(6x^2 + 5)\). Here, we have terms like \(18x^3 + 15x\) and \(-12x^2 - 10\) emerging from multiplication.
Understanding polynomial expressions is crucial because they allow us to understand complex relationships and solve equations in algebra and beyond. Always look for patterns or common terms as they can frequently lead to quicker simplifications.
Factoring Techniques
Factoring is a technique used to simplify polynomial expressions by expressing them as a product of factors. It involves identifying components that can be multiplied together to produce the original polynomial. Here鈥檚 how it works:
  • Firstly, identify any common factors, as shown in the exercise problem with \((6x^2 + 5)\). Notice how this common component simplifies the whole expression.
  • Once identified, "factor out" this common term to rewrite the polynomial in a simpler form: \((6x^2 + 5)(3x - 2)\).
  • Finally, always verify your work by expanding the factored expression to ensure it matches the original polynomial. For instance, expanding \((6x^2 + 5)(3x - 2)\) verifies it simplifies back to \(3x(6x^2 + 5) - 2(6x^2 + 5)\).
Using factoring techniques not only simplifies solving but also aids in understanding deeper mathematical structures. Remember, practice helps in mastering how to spot potential common factors and rewrite polynomials effectively.

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Most popular questions from this chapter

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}-36 $$

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}+8 x z+7 z^{2} $$

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}-15 x+54 $$

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7\) find the following. \(-5[P(x)]-Q(x)\)

Explain how to convert a number from standard notation to scientific notation.
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