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Chapter 9: Problem 8

Find the least common multiple of each pair of polynomials. \(5 y^{2}-80\) and \(y+4\)

Short Answer

Expert verified
The least common multiple of the pair of polynomials \(5 y^{2}-80\) and \(y+4\) is \(5(y+4)(y-4)\)

Step by step solution

01

Factorize the Polynomial Expressions

To begin with, let's factorize each of the given polynomial expressions. Factoring a polynomial process involves breaking the polynomial down into simpler terms that when multiplied, gives the original polynomial. Factorize \(5 y^{2}-80\) and \(y+4\).
02

Factorization of \(5 y^{2}-80\)

Recognize that \(5 y^{2}-80\) is a difference of two squares which can be factorized into the form \((a+b)(a-b)\). Therefore, it can be factorized as \(5(y^{2}-16)\) and further more as \(5(y+4)(y-4)\).
03

Factorization of \(y+4\)

\The polynomial \(y+4\) is already in its simplest form and cannot be further factorized.
04

Determine the Least Common Multiple (LCM)

The LCM of the polynomials is found by multiplying each factor the greatest number of times it occurs in either polynomial. After factoring, it's seen that both polynomials have a common factor of \(y+4\). Therefore, the LCM is \(5(y+4)(y-4)\).

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is like breaking down a number into its prime components, making it easier to work with. When we factor a polynomial, we're looking for expressions that multiply together to get the original polynomial. This process involves identifying terms or expressions that can be multiplied to reconstruct the original equation.
For example, consider the polynomial expression given in the exercise:
  • The polynomial is \(5y^2 - 80\). At first glance, it seems complex, but it can be simplified by factoring.
  • Notice how we can pull out a common factor or recognize common patterns like the difference of squares.

Breaking down the polynomial into \(5(y^2 - 16)\), leads us to a simpler expression that can be further analyzed using other factoring methods like the difference of squares.
Difference of Squares
The difference of squares is a special factoring technique that deals with expressions of the form \(a^2 - b^2\). This concept allows you to factor expressions using the pattern \((a+b)(a-b)\). It works when you see a subtraction (difference) between two perfect squares.
In the problem, the expression can be arranged as:
  • The key is to identify each square component. For \(y^2 - 16\), notice that both \(y^2\) and \(16\) are perfect squares.
  • \(16\) can be rewritten as \(4^2\).
  • Thus, you apply the pattern, obtaining: \((y+4)(y-4)\).
Breaking down into these factors simplifies the polynomial and helps identify common factors crucial for the least common multiple of polynomials.
Greatest Common Factor
The greatest common factor (GCF) is the highest factor that divides each of the numbers or polynomials involved. Finding the GCF helps simplify expressions by identifying shared factors.
In the original problem, this procedure is critical to finding the least common multiple (LCM). Here's how you can think about it:
  • The factor \(y+4\) is present in both polynomial expressions: \(5(y+4)(y-4)\) and \(y+4\).
  • Since \(y+4\) is common, it's a shared factor, meaning it's part of the GCF for the two polynomials.
Finding and acknowledging the GCF simplifies the exercise of determining the least common multiple, eventually revealing that the LCM of the pair of polynomials is \(5(y+4)(y-4)\). This step ensures you cover all factors appropriately without redundancy.

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Most popular questions from this chapter

Use this information for Exercises \(53-58\) . Bag 1 contains 5 red marbles, 1 blue marble, 3 yellow marbles, and 2 green marbles. Bag 2 contains 1 red pencil, 3 red pens, 2 blue pencils, and 5 blue pens. One item is drawn from bag \(2 .\) What is the probability that it is red?

Two number cubes are rolled. What is the probability that the sum is greater than 9 or less than 6\(?\)

What are the restrictions on \(x\) when \(\frac{x^{2}-x-2}{x^{2}-9}\) is divided by \(\frac{x-8}{x^{2}+10 x+25} ?\) $$ \begin{array}{ll}{F . x \neq-3 \text { or }-5} & {\text { G. } x \neq 3,-3, \text { or }-5} \\ {\text { H. } x \neq 3,-3,-5, \text { or } 8} & {\text { J. } x \neq 2,9,8, \text { or }-25}\end{array} $$

What is the product of \(\frac{4 x^{2}-1}{2 x^{2}-5 x-3}\) and \(\frac{x^{2}-6 x+9}{2 x^{2}+5 x-3} ?\) $$ \begin{array}{ll}{\text { A. } 1} & {\text { B. } x-3} \\ {\text { C. } x+3} & {\text { D. } \frac{x-3}{x+3}}\end{array} $$

Let \(f(x)=x^{2}+1\) and \(g(x)=3 x .\) Find each value. \((g \circ f)\left(\frac{1}{2}\right)\)
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