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Chapter 5: Problem 53

Factor each polynomial by factoring out the GCF. $$4 b^{3}-10 b^{2}$$

Short Answer

Expert verified
The factorized form of the polynomial \(4 b^{3}-10 b^{2}\) by factoring out the GCF is \(2 b^{2}(2b - 5)\).

Step by step solution

01

Identify the GCF

First, determine the greatest common factor. Here the two terms are \(4 b^{3}\) and \(-10 b^{2}\). Here numerical GCF is 2 and the highest power of b that is common to both terms is \(b^2\). Thus, the GCF is \(2 b^{2}\).
02

Factor out the GCF

Factor the GCF from the original expression. Divide each term in the original polynomial by the GCF and write the results inside parentheses. For \(4 b^{3}\), this results in \(2b\) and for \(-10 b^{2}\), this results in -5. So, after factoring, the expression becomes \(2 b^{2}(2b - 5)\).

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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 a key concept in factoring and is the largest factor that divides two or more numbers. By finding the GCF, we can simplify polynomial expressions effectively.

To find the GCF of polynomial terms, follow these steps:
  • Identify the numerical coefficients of each term.
  • Determine the numerical GCF by finding the highest number that divides each coefficient.
  • Next, look at the variables and their exponents. Identify the common variables with the lowest exponent between the terms.
By combining these, you determine the overall GCF. In our example, the terms were \( 4b^3 \) and \(-10b^2\). The numerical GCF is 2, and the variable part is \( b^2 \). Thus, the GCF is \( 2b^2 \).
Polynomial Expressions
Polynomial expressions consist of terms, which are combinations of constants, variables, and exponents. These expressions can be classified based on the number of terms they contain.

Polynomials are crucial in algebra since they help describe a wide range of mathematical scenarios. Here are some key points about polynomial expressions:
  • Each term in a polynomial is separated by a plus or minus sign.
  • A monomial has one term, a binomial has two, and a trinomial has three.
  • The highest exponent in a polynomial determines its degree.
Understanding polynomial expressions is essential for carrying out algebraic operations such as addition, subtraction, and especially factoring.
Algebraic Factoring
Algebraic factoring is a technique used to simplify polynomials by breaking them down into more manageable pieces, or 'factors'. Factoring is like finding a puzzle's pieces before putting them together.

In algebraic factoring, we follow these general steps:
  • Identify the GCF of all terms in the polynomial.
  • Divide each term by the GCF, resulting in a simpler expression inside the parentheses.
  • Write the factor (the GCF) multiplied by this simpler expression.
Factoring polynomials helps solve equations and is crucial for calculus and other advanced mathematics. In our exercise, factoring out \( 2b^2 \) from \( 4b^3 - 10b^2 \) simplified the expression to \( 2b^2(2b - 5) \). Recognizing these patterns allows more complex problems to be approached with confidence.

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Most popular questions from this chapter

Factor. If the polynomial is prime, so indicate. $$30 r^{5}+63 r^{4}-30 r^{3}$$

Factor. If the polynomial is prime, so indicate. $$9 x^{2}+21 x-18$$

Factor. $$9-a^{2}-4 a b-4 b^{2}$$

Write the terms of each trinomial in descending powers of one variable. Then factor. $$12 x^{2}+10 y^{2}-23 x y$$

Factor. $$z^{2}-2 z+1$$
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