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Chapter 6: Problem 27

Find the least common multiple of each pair of polynomials. $$ x^{2}-9, x^{2}-6 x+9 $$

Short Answer

Expert verified
The LCM is \((x+3)(x-3)^2\).

Step by step solution

01

Factor both Polynomials

Start by factoring each polynomial. For the first polynomial, \(x^2 - 9\), apply the difference of squares formula: \(x^2 - 9 = (x+3)(x-3)\). Next, factor the second polynomial, \(x^2 - 6x + 9\). Notice it is a perfect square trinomial: \(x^2 - 6x + 9 = (x-3)(x-3)\) or \((x-3)^2\).
02

Determine the Least Common Multiple (LCM)

To find the least common multiple, identify the unique factors and use the highest power of each factor present in the factorizations. Combining the factors from both polynomials, \((x+3)(x-3)\) and \((x-3)^2\), we take the highest power of \(x-3\) which is \((x-3)^2\) and include \(x+3\) once to get: \((x+3)(x-3)^2\).
03

Simplify the LCM

Thus, the LCM of \(x^2 - 9\) and \(x^2 - 6x + 9\) is \((x+3)(x-3)^2\).

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Key Concepts

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

Factoring Polynomials
When we factor polynomials, we are breaking them down into simpler polynomials that, when multiplied together, give the original polynomial. This process is crucial in algebra and helps solve many mathematical problems. For example, to factor the polynomial \(x^2 - 9\), we recognize it as a difference of squares. The formula for difference of squares is \(a^2 - b^2 = (a + b)(a - b)\). Here, \(x^2 - 9\) can be written as \(x^2 - 3^2\), allowing us to factor it as \( (x + 3)(x - 3)\). Developing the skill to quickly identify and factor such forms simplifies solving polynomial equations and finding least common multiples.
Difference of Squares
The difference of squares is a particular type of polynomial that fits the pattern \(a^2 - b^2\). It can be factored into \( (a + b)(a - b)\). Recognizing this form is very useful for factoring polynomials. In our example, \(x^2 - 9\) fits this pattern because it can be rewritten as \(x^2 - 3^2\). Applying the formula, we factor it into \( (x + 3)(x - 3)\). This is a straightforward process once you get the hang of spotting squares and their differences. The difference of squares is a fundamental tool for simplifying polynomials and finding their factors.
Perfect Square Trinomials
A perfect square trinomial is a special form of polynomial that can be written as the square of a binomial. It usually appears in the form \(a^2 \pm 2ab + b^2\), which factors into \( (a \pm b)^2\). For the polynomial in our exercise, \(x^2 - 6x + 9\), we identify it as a perfect square trinomial because it follows the pattern of \(x^2 - 2 \cdot 3 \cdot x + 3^2\). This trinomial can be factored down to \( (x - 3)^2\). Recognizing perfect square trinomials simplifies factoring and further operations like finding the least common multiple.
Least Common Multiple
To find the least common multiple (LCM) of polynomials, we need to identify the highest power of all unique factors from the given polynomials. For instance, in our example: we have the factored forms \( (x + 3)(x - 3) \) and \( (x - 3)^2\). Here, the common factors are \(x + 3\) and \(x - 3\). The highest powers present are \( (x + 3)^1 \) and \( (x - 3)^2 \). So, the LCM is given by \( (x + 3)(x - 3)^2 \). Finding the LCM helps in simplifying expressions and solving polynomial equations that involve multiple terms.

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Most popular questions from this chapter

For each pair of functions fand \(g\), find all values of a for which \(f(a)=g(a)\) $$ f(x)=\frac{1}{1+x}+\frac{x}{1-x}, g(x)=\frac{1}{1-x}-\frac{x}{1+x} $$

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