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Magazine Chapter 0: Problem 30
Use grouping to factor the polynomial. \(2 x^{4}-5 x^{3}+10 x-25\)
Short Answer
Expert verified
The polynomial factors into \((x^3 + 5)(2x - 5)\).
Step by step solution
01
Group the Terms
We begin by grouping the terms in pairs to explore the possibility of factoring by grouping. Group the polynomial into two parts: \((2x^4 - 5x^3) + (10x - 25)\).
02
Factor Out the Greatest Common Factor (GCF)
Look for the greatest common factor in each group. For the first group, \(2x^4 - 5x^3\), the GCF is \(x^3\), and for the second group, \(10x - 25\), the GCF is \(5\). Factor each group:- From \(2x^4 - 5x^3\), factor out \(x^3\) to get \(x^3(2x - 5)\).- From \(10x - 25\), factor out \(5\) to get \(5(2x - 5)\).
03
Identify the Common Binomial
After factoring both groups, we have the expression:\(x^3(2x - 5) + 5(2x - 5)\).Notice that \((2x - 5)\) is a common binomial factor in both terms.
04
Factor Out the Common Binomial
Since \((2x - 5)\) is common to both terms, factor it out:\((x^3 + 5)(2x - 5)\).
05
Verify the Factorization
Double-check the factorization by expanding \((x^3 + 5)(2x - 5)\) to ensure that it results in the original polynomial. Expand using the distributive property:- First, \(x^3 \times 2x = 2x^4\).- Then, \(x^3 \times (-5) = -5x^3\).- Next, \(5 \times 2x = 10x\).- Finally, \(5 \times (-5) = -25\).Thus, the original polynomial \(2x^4 - 5x^3 + 10x - 25\) is indeed equivalent to \((x^3 + 5)(2x - 5)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Grouping Method
When you encounter a polynomial that seems complex, the grouping method offers a structured way to simplify it. It involves organizing terms into pairs or groups that can be factored. Start by looking at a polynomial's terms and splitting them into two parts. Here is an example: for the polynomial \(2x^4 - 5x^3 + 10x - 25\), divide it into
- \((2x^4 - 5x^3)\): The first group
- \((10x - 25)\): The second group
- Look for common factors within each group.
- Re-factor to uncover any repeated patterns or terms.
- Often, this technique reveals a common binomial factor, simplifying the expression.
Greatest Common Factor
The Greatest Common Factor (GCF) is a crucial concept for simplifying expressions. It's the largest factor that divides two or more terms without leaving a remainder. By identifying and factoring out the GCF, you reduce the polynomial into simpler, smaller forms.Steps to Find the GCF:
- Examine each term in the group separately.
- For \(2x^4 - 5x^3\), the GCF is \(x^3\).
- For \(10x - 25\), the GCF is \(5\).
Binomial Factor
A binomial factor is a two-term expression that appears throughout a polynomial. Recognizing and factoring out a binomial can drastically simplify the polynomial's form. Once the terms are grouped and the GCF is extracted, look for similar terms within those results.Example:
- In \((x^3(2x - 5) + 5(2x - 5))\), \((2x-5)\) is a common binomial factor.
Distributive Property
The distributive property is a fundamental aspect of algebra, enabling the multiplication of an entire expression by a single term. This property aids in expanding or simplifying expressions. Verification:
- Use it to check if factoring was done correctly.
- Multiply each term: \(x^3 \times 2x = 2x^4\) and \(x^3 \times (-5) = -5x^3\).
- Similarly, \(5 \times 2x = 10x\) and \(5 \times (-5) = -25\).
