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Chapter 6: Problem 30

For Problems \(11-46\), factor each polynomial completely. (Objective 2) $$ 84 x^{2} y^{3}+12 x y^{3} $$

Short Answer

Expert verified
The polynomial \(84x^2y^3 + 12xy^3\) factors completely to \(12xy^3(7x + 1)\).

Step by step solution

01

Identify Common Factors

The given expression is \(84x^2y^3 + 12xy^3\). Both terms contain \(xy^3\) as a factor. Identify this common factor to factor out.
02

Factor Out the Common Factor

Factor \(12xy^3\) out of each term: \[ 84x^2y^3 + 12xy^3 = xy^3(84x + 12) \] Now, we have the expression \(xy^3(84x + 12)\).
03

Simplify the Remaining Expression

The expression inside the parentheses \( (84x + 12) \) can be further simplified by factoring out the greatest common divisor of 84 and 12, which is 12.
04

Perform Final Factoring

Factor 12 out of \( (84x + 12) \): \[ 84x + 12 = 12(7x + 1) \]Now rewrite the entire expression as: \[ xy^3(12)(7x + 1) \]
05

Express the Total Factorization

The fully factored expression is:\[ 12xy^3(7x + 1) \]

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Key Concepts

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

Common Factors
When working with polynomials, a common step in simplification is to look for common factors. In the expression \(84x^2y^3 + 12xy^3\), you'll notice that both terms share some elements:
  • They both have the variable \(x\).
  • They both have \(y^3\) as a component.
A common factor is a term that divides each of the components in the polynomial without leaving a remainder. When you identify a common factor, you can pull it out (or "factor it out") to simplify the expression. In our example, the entire \(xy^3\) expression is common in each term. By factoring \(xy^3\) out of the expression, you ensure that you are simplifying efficiently. This step is foundational in making the polynomial easier to work with and paves the way for further simplification.
Greatest Common Divisor
After identifying and factoring the common factors, the next step often involves working with numerical coefficients. In the expression \((84x + 12)\), we need to find the greatest common divisor (GCD) of 84 and 12. The GCD is the largest number that divides both numbers exactly.To find the GCD, you can use either:
  • Prime factorization: Breaking down each number into its prime factors.
  • Euclidean algorithm: A more systematic approach for larger numbers.
For example, using prime factorization:
  • 84 can be expressed as \(2^2 \times 3 \times 7\).
  • 12 can be expressed as \(2^2 \times 3\).
The GCD is \(2^2 \times 3 = 12\). This means we can further factor the expression to \(12(7x + 1)\), simplifying it significantly. Always remember that handling numerical coefficients carefully is essential for accurate and complete polynomial simplification.
Polynomial Simplification
Polynomial simplification is the process of breaking down a polynomial into a product of simpler polynomials or numbers. This process involves several steps, including factoring out common factors and using the greatest common divisor (GCD). For our polynomial \(84x^2y^3 + 12xy^3\), simplification means rewriting it to its most compact form.Here's the step-by-step:
  • Factor out the common term \(xy^3\), which results in \(xy^3(84x + 12)\).
  • Inside the parentheses, \(84x + 12\), identify 12 as the GCD of the numerical coefficients and factor it out, getting \(12(7x + 1)\).
  • Combine these results to reach the fully factored expression: \(12xy^3(7x + 1)\).
Simplification not only makes expressions easier to understand but also prepares polynomials for further operations or integration with other expressions. By reducing expressions to their simplest forms, you strengthen your understanding and manipulation skills in algebra.

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Most popular questions from this chapter

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ n^{2}-28 n+192=0 $$

What does the expression "not factorable using integers" mean to you?

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) A strip of uniform width is to be cut off of both sides and both ends of a sheet of paper that is 8 inches by 11 inches, in order to reduce the size of the paper to an area of 40 square inches. Find the width of the strip.

Find two consecutive odd whole numbers whose product is 63 .

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ 25 x^{2}+60 x+36=0 $$
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