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Polynomial identities are effective tools for making our work with polynomials much easier. In this lesson, we will define polynomial identities and explore some useful examples.
Sometimes you run across polynomial problems that defy rationality. They just stare at you, daring you to think of an approach. There seems to be no intuitive way to solve the problem. In many of these situations, a polynomial identity has been created that will burst open the door and make the problem easy. In this lesson, we're going to explore some of these magic keys that can make all the difference between a seemingly impossible problem and an easy one.
Technically, polynomial identities are just equations that are true, but identities are particularly useful for showing the relationship between two seemingly unrelated expressions. Since the identity lets you make that connection, you can then use it to convert back and forth between the two expressions, grabbing the side that happens to be useful at the moment.
Some Useful Identities
There are many popular polynomial identities in the math world, and here are some valuable ones:
(a + b)² = a² + 2ab + b²
This one can speed up your factoring and FOIL (First - Outside - Inside - Last) multiplying. When a binomial is squared, it always breaks down to the same expression. A similar identity is the one where the terms are being subtracted:
(a - b)² = a² - 2ab + b²
When you see a polynomial in either form on the right side of the identity, you know that it will factor into the expression on the left. Remember, these are not the same as a² + b², which doesn't have an identity for factoring.
Difference Between Squares
The difference between squares identity can save you many hours of factoring and multiplying:
a² - b² = (a + b) (a - b)
This identity is so useful you find yourself looking hopefully through your polynomial problems, delighted when you see any form of a difference between squares. Once again, the sum of squares a² + b², won't factor (at least, not into real numbers) and doesn't have a useful identity like these.
Sum/Difference of Cubes
Life savers in terms of polynomial factoring are the sum of cubes and difference of cubes identities. These are some of the least intuitive relationships in your toolbox, and are extremely useful to remember.
a³ + b³ = (a + b) (a² - ab + b²) is the sum of cubes and a³ - b³ = (a - b) (a² + ab + b²) is the difference of cubes.
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Although it's not all that fun to try to remember, the quadratic formula is one of your most useful polynomial identities. It allows you to break down any quadratic expression (polynomial with a squared term as its highest order expression) into its factors:
Figure 1: Quadratic Formula
If ax² + bx + c = 0 then you can use the formula in that we just covered to find the factors (real or imaginary). Notice that a, b, and c represent the coefficients (as in, the numeric part of the terms) that appear in the quadratic equation. When you use this identity to solve a quadratic equation, you may end up with one or two real values for x, or you might end up trying to take the square root of a negative number, which means you'll end up with imaginary results!
Here are a couple of common polynomial identities that might be useful while you're wandering in the algebraic minefields, especially working with quadratic equations:
(a + b) (c + d) = ac + ad + bc + bd (the FOIL formula)
x² + (a + b) x + ab = (x + a)(x + b) (another FOIL formula)
Let's take a couple of moments to review what we've learned. Polynomial identities are equations that are true and have general applications in your algebra work. They're tools for moving more quickly through homework and other math activities. The more familiar you are with these, the easier your algebra work will be! Here are just a few:
(a + b)² = a² + 2ab + b², or (a - b)² = a² - 2ab + b²
The difference between squares identity: a² - b² = (a + b) (a - b)
The sum of cubes identity: a³ + b³ = (a + b) (a² - ab + b²)
The difference of cubesidentity: a³ - b³ = (a - b) (a² + ab + b²)
The FOIL formula: (a + b) (c + d) = ac + ad + bc + bd
Another FOIL formula: x² + (a + b) x + ab = (x + a)(x + b)
The quadratic formula, which allows you to break down any quadratic expression into its factors.
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