Lesson 24

Polynomial Identities (Part 2)

• Let’s explore some other identities.

24.1: Revisiting an Old Theorem

Instructions to make a right triangle:

• Choose two integers.
• Make one side length equal to the sum of the squares of the two integers.
• Make one side length equal to the difference of the squares of the two integers.
• Make one side length equal to twice the product of the two integers.

Follow these instructions to make a few different triangles. Do you think the instructions always produce a right triangle? Be prepared to explain your reasoning.

24.2: Theorems and Identities

Here are the instructions to make a right triangle from earlier:

• Choose two integers.
• Make one side length equal to the sum of the squares of the two integers.
• Make one side length equal to the difference of the squares of the two integers.
• Make one side length equal to twice the product of the two integers.
1. Using $$a$$ and $$b$$ for the two integers, write expressions for the three side lengths.
2. Why do these instructions make a right triangle?

24.3: Identifying Identities

Here is a list of equations. Circle all the equations that are identities. Be prepared to explain your reasoning.

1. $$a = \text-a$$
2. $$a^2+2ab+b^2=(a+b)^2$$
3. $$a^2-2ab+b^2=(a-b)^2$$
4. $$a^2-b^2 = (a-b)(a-b)$$
5. $$(a+b)(a^2-ab+b^2)=a^3-b^3$$
6. $$(a-b)^3=a^3-b^3-3ab(a+b)$$
7. $$a^2(a-b)^4-b^2(a-b)^4 = (a-b)^5(a+b)$$

24.4: Egyptian Fractions

In Ancient Egypt, all non-unit fractions were represented as a sum of distinct unit fractions. For example, $$\frac49$$ would have been written as $$\frac13 + \frac19$$ (and not as $$\frac19+\frac19+\frac19+\frac19$$ or any other form with the same unit fraction used more than once). Let’s look at some different ways we can rewrite $$\frac{2}{15}$$ as the sum of distinct unit fractions.

1. Use the formula $$\frac{2}{d}=\frac{1}{d}+\frac{1}{2d}+\frac{1}{3d}+\frac{1}{6d}$$ to rewrite the fraction $$\frac{2}{15}$$, then show that this formula is an identity.
2. Another way to rewrite fractions of the form $$\frac{2}{d}$$ is given by the identity $$\frac{2}{d} = \frac{1}{d} +\frac{1}{d+1} + \frac{1}{d(d+1)}$$. Use it to re-write the fraction $$\frac{2}{15}$$, then show that it is an identity.

For fractions of the form $$\frac{2}{pq}$$, that is, fractions with a denominator that is the product of two positive integers, the following formula can also be used: $$\frac{2}{pq}=\frac{1}{pr} + \frac{1}{qr}$$, where $$r = \frac{p+q}{2}$$. Use it to re-write the fraction $$\frac{2}{45}$$, then show that it is an identity.

Summary

Sometimes we can think something is an identity when it actually isn’t. Consider the following equations that are sometimes mistaken as identities:

$$\displaystyle (a+b)^2=a^2+b^2$$

$$\displaystyle (a-b)^2=a^2-b^2$$

Both of these are true for some very specific values of $$a$$ and $$b$$, for example when either $$a$$ or $$b$$ is 0, but they are not true for most values of a and b, for example $$a = 2$$ and $$b = 1$$ (try it!). The actual identities associated with the expressions on the left side are $$(a+b)^2=a^2+2ab+b^2$$ and $$(a-b)^2=a^2-2ab+b^2$$.

Are polynomials the only types of expressions you can find in identities? Not at all! Here is an identity that shows a relationship between rational expressions:

$$\displaystyle \frac{1}{x}=\frac{1}{x+1} + \frac{1}{x(x+1)}$$

We can show that this identity is true by adding the terms in the expression on the right using a common denominator:

$$\displaystyle \begin{array} \\ \frac{1}{x+1} + \frac{1}{x(x+1)} &= \frac{1}{x+1} \boldcdot \frac{x}{x} + \frac{1}{x(x+1)} \\ &= \frac{x}{x(x+1)} + \frac{1}{x(x+1)} \\ &= \frac{x + 1}{x(x+1)} \\ &= \frac{1}{x} \\ \end{array}$$

An important difference from polynomial identities is that identities involving rational expressions could have a few exceptional values of $$x$$ where they are not true because the rational expressions on one side or the other are not defined. For example, the identity above is true for all values of $$x$$ except $$x=0$$ and $$x =\text-1$$.

Glossary Entries

• identity

An equation which is true for all values of the variables in it.