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Chapter 6: Problem 17

Factor the given expressions completely. $$x^{6} y^{3}+x^{3} y^{6}$$

Short Answer

Expert verified
\(x^6 y^3 + x^3 y^6\) factors to \(x^3 y^3 (x + y)(x^2 - xy + y^2)\).

Step by step solution

01

Identify Common Factors

Look for common factors between the terms in the expression, \(x^6 y^3 + x^3 y^6\). Notice that each term has at least \(x^3 y^3\) as a common factor. Factoring \(x^3 y^3\) out, we get:\[x^3 y^3(x^3 + y^3) \]
02

Recognize a Sum of Cubes

The expression \(x^3 + y^3\) is a sum of cubes, which can be factored using the formula for the sum of cubes: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). In this case, \(a = x\) and \(b = y\).
03

Apply the Sum of Cubes Formula

Factor \(x^3 + y^3\) using the sum of cubes formula:\[x^3 + y^3 = (x + y)(x^2 - xy + y^2)\]
04

Write the Complete Factored Expression

Substitute the factored form of \(x^3 + y^3\) back into the expression. Thus, the completely factored form of \(x^6 y^3 + x^3 y^6\) is:\[x^3 y^3 (x + y)(x^2 - xy + y^2)\]

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Key Concepts

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

Sum of Cubes
Factoring expressions often involves recognizing special patterns like the sum of cubes. The expression \(x^3 + y^3\) is such a pattern. When you see a cube term added to another cube, you're looking at a sum of cubes. This pattern can be tricky to spot but it's simple once you remember its distinctive factoring formula: \[a^3 + b^3 = (a+b)(a^2 - ab + b^2)\] Here, \(a\) and \(b\) represent the base terms of the cubes.
  • Identify the Cubes: In \(x^3 + y^3\), \(x\) and \(y\) are the bases.
  • Apply the Formula: Use \((x+y)(x^2 - xy + y^2)\) to factor it.
Remember, this formula only applies to sums, not differences of cubes, which have their own formula. Getting comfortable with these formulas will make factoring much easier in algebra.
Common Factors
One of the first and most important steps in polynomial factorization is identifying the common factors. It simplifies the expression significantly by reducing it to smaller components. When faced with an expression like \(x^6 y^3 + x^3 y^6\), look for the greatest quantity that divides all terms without a remainder.
  • Inspect Each Term: Both terms share multiplicands of \(x^3\) and \(y^3\).
  • Factor Out the Common Part: By extracting \(x^3y^3\) from each term, the expression becomes simpler.
This can be visualized as taking the common part outside a bracket, like unwrapping a gift to make the contents (in this case, our expression) easier to manage.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler polynomials that, when multiplied together, give you back the original polynomial. It's like unscrewing a machine to see its parts.

In this exercise, we had to factor \(x^6 y^3 + x^3 y^6\) completely. Here's a breakdown of how polynomial factorization works:
  • Find Common Factors First: This step simplifies the expression to a more recognizable form.
  • Identify Special Patterns: Recognize expressions like sums of cubes or differences of cubes to apply the correct formula.
Breaking down complex polynomial expressions into manageable parts can be likened to solving a puzzle. As you gain more practice, it will become intuitive to see which factors to extract and which formulas to apply.

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Most popular questions from this chapter

Solve the given equations and check the results. $$\frac{1}{x^{2}-x}-\frac{1}{x}+\frac{1}{1-x}=0$$

After finding the simplest form of each fraction, explain why it cannot be simplified more. (a) \(\frac{x^{3}-x}{1-x}\) (b) \(\frac{2 x^{2}+4 x}{2 x^{2}+4}\)

Find the required expressions. A boat that travels at \(v \mathrm{km} / \mathrm{h}\) in still water travels \(5 \mathrm{km}\) upstream and then \(5 \mathrm{km}\) downstream in a stream that flows at \(w \mathrm{km} / \mathrm{h}\). Write an expression for the total time the boat traveled as a single fraction.

Solve for the indicated letter. Each of the given formulas arises in the technical or scientific area of study listed. \(\frac{1}{C}=\frac{1}{C_{2}}+\frac{1}{C_{1}+C_{3}},\) for \(C_{1} \quad\) (electricity: capacitors)

Solve the given equations and check the results. $$\frac{6}{4 x-6}+\frac{2}{4}=\frac{6}{3-2 x}$$
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