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Chapter 6: Problem 28

Find the least common multiple of each pair of polynomials. $$ x^{2}-x-12, x^{2}-16 $$

Short Answer

Expert verified
\[ \text{LCM} = (x - 4)(x + 3)(x + 4) \]

Step by step solution

01

Factorize the first polynomial

Given polynomial: \[ x^{2} - x - 12 \] Factorize it into two binomials: \[ x^{2} - x - 12 = (x - 4)(x + 3) \]
02

Factorize the second polynomial

Given polynomial: \[ x^{2} - 16 \] Factorize it as a difference of squares: \[ x^{2} - 16 = (x - 4)(x + 4) \]
03

Determine the least common multiple

Find the least common multiple (LCM) of the factored forms. The LCM should include each unique factor the maximum number of times it occurs in any one of the polynomials: \[ (x - 4), (x + 3), \text{ and } (x + 4) \] Thus, the LCM is: \[ (x - 4)(x + 3)(x + 4) \]
04

Write the final expression

Combine the factors to write the final expression for the LCM: \[ \text{LCM} = (x - 4)(x + 3)(x + 4) \]

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Key Concepts

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

Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into products of simpler polynomials. This step-by-step process helps in simplifying complex expressions and solving polynomial equations easily. For instance, in the exercise, we factored the polynomial \(x^2 - x - 12\) into two binomials as follows:
\(x^2 - x - 12 = (x - 4)(x + 3)\).
To factorize, you need to find two numbers that multiply to give the constant term and add up to give the coefficient of the middle term. Here:
  • Multiply to: -12
  • Add up to: -1
These numbers are -4 and 3.
Once identified, express the original polynomial as a product of two binomials using these numbers.
Difference of Squares
The difference of squares is a specific type of polynomial factoring used when the polynomial is in the form \(a^2 - b^2\). This can be rewritten as \((a - b)(a + b)\).
In the given exercise, for the polynomial \(x^2 - 16\), recognize it follows this form because:
鈥� \(x^2 = a^2\)
鈥� \(16 = 4^2\)
Thus, you can factorize it using the difference of squares rule:
\(x^2 - 16 = (x - 4)(x + 4)\).
This method is a powerful tool for simplifying polynomial expressions and helps in finding the least common multiple (LCM).
Binomials
A binomial is a polynomial with exactly two terms. In the context of factoring, understanding binomials is crucial. For example, in the factorization step, both binomials derived from the original polynomials are used:
  • \((x - 4)(x + 3)\)
  • \((x - 4)(x + 4)\)
To find the LCM of these polynomials, identify the unique binomial factors and include each one the maximum number of times it appears in any factorization. Hence, the LCM in the given problem is formed by:
  • Common factor from both: \((x - 4)\)
  • Remaining factors: \((x + 3)\) and \((x + 4)\)
This results in the final LCM being: \((x - 4)(x + 3)(x + 4)\). Understanding binomials simplifies the process of factoring and finding LCM.

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Most popular questions from this chapter

The intensity I of light from a bulb varies directly as the wattage of the bulb and inversely as the square of the distance \(d\) from the bulb. If the wattage of a light source and its distance from reading matter are both doubled, how does the intensity change?

Find the domain of \(f\). \(f(x)=\frac{3 x}{x^{2}+1}\)

Find an equation of variation in which: \(y\) varies inversely as the square of \(x,\) and \(y=50\) when \(x=10\).

Find the variation constant and an equation of variation if y varies directly as \(x\) and the following conditions apply. \(y=2\) when \(x=\frac{1}{5}\)

Find and simplify $$ \frac{f(x+h)-f(x)}{h} $$ for each rational function \(f\) $$ f(x)=\frac{3}{x} $$
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