- Early usage: For LarrySanger
At the high school level quadratic equations are useful in displaying the teacher's facility in proving the quadratic formula, by completing the square. ---- No thanks necessary.----
Questions (I'd rather not make changes to the page since it's out of my area of expertise):
- Can I have more examples please? See below.
- What are polynomials good for? Describing some kinds of relations between variables. Like much of math, we present definitions and techniques and leave the applications to others.
- Maybe an example of a high school textbook problem involving polynomials would help orient me...just a suggestion, feel totally free to ignore it.
Further examples of polynomials (some are monomials which form a special case with only one term):
- Area of a square = side2
- Volume of a cube = side3.
- Area of a square with its lower left corner at the point (x,x) and its upper right corner at (y,y) = (y-x)2 = y2-2xy+x2.
- This is a polynomial in the two variables, x and y.
If somebody wants to integrate my writeup on E2 to here, feel free. The AC Method may be of particular interest. This is primarily just telling how to factor polynomials so there might be a better place (i.e. factoring) to put it. For simplicity, I'll post a partially wikified version here. If you think it's useful, integrate it. Else, just remove it: http://everything2.org/?node_id=895118 (Note: could contain some errors.)
anxn + an-1xn-1 + an-2xn-2. . . a1x + a0
The degree of a polynomial is the highest total of powers of variables (x, y, etc.) of a single term, so in the polynomial 2xy2 + x2 the degree is three (in the first term, x has a power of one). The standard form of a polynomial is when you write it with the degrees descending (x2 + x + 3, not x + x2 + 3)
Ex 1) 2x^2 + 4x = 2x(x + 2)
- Ex 2) 2x2 + 6x + 8 = 2(x2 + 3x + 4)
You usually have to find two binomials (B1 and B2) whose first terms multiply to the first term of your trinomial, last terms multiply to the last term of the trinomial, and B1's first term times B2's last term plus vise versa equals the middle term (FOIL users: Inside + Outside=Middle)
Ex 3) x2 + 3x + 2 = (x + 1)(x + 2)
If the first term of your trinomial has a coefficient (a) of 1--as shown above--then the first terms of the binomials are x. Otherwise, you have to play around searching for the proper factors to get it right. That's where the following method comes in:
The AC Method
First factor out the common factor. Always, always, always do this.
- Now you have ax2 + bx + c a isn't 1.
- Change it to x2 + bx + ac. (If you're stuck wondering how the hell to move the a all the way over to the c, don't bother. Just do it.)
- Factor x2 + bx + ac into your (presumably) two binomials. Then stick a back into the first terms of both of them, factor out the common factor and toss it out. You're done.
Ex 4) 6x2 + 2x-4
- 2(3x2 + x-2) (Factor out common factor)
- 2(x2 + x-6) (move a to third term)
- 2(x + 3)(x-2) (factor)
- 2(3x + 3)(3x-2) (put a back into first terms)
- 2(x + 1)(3x-2) (factor out and delete common factor)
- If you're planning on using the AC Method a lot you may want to work on your factoring large numbers because ac is often rather large.
Now, I know you're thinking, "What if I have a four-term (or more) polynomial?" Easy: Take a few terms, and slap parenthesis around them (Hint, put together terms that have common factors or that look like they'll factor easily.)
Ex 5) 2x3 - 3x2 + 4x - 6
- (2x3 - 3x2) + (4x - 6)
- x2(2x - 3) + 2(2x - 3)
- (x2 + 2)(2x - 3)
That last example (first and last steps anyway) was taken from College Algebra by Michael Sullivan because I was having a heck of a time making up a good example. (I'm always coming up with prime polynomials in my example and having to modify them so I can factor them. I wish my math teacher had let me do that in my homework.)
Difference of Squares: x2 - a2 = (x - a)(x + a) (Ex 6) x2 - 144 = (x + 12)(x - 12))
- Perfect Squares: x2 ± 2ax + a2 = (x ± a)2
- Unnamed, but bears remembering: x2 (a + b)x + ab = (x + a)(x + b)
- Unnamed, but bears remembering: acx2 + (ad + bc)x + bd = (ax + b)(cx + d)
- Perfect Cubes: x3 + 3ax2 + 3a2x + a3 = (a + x)3, x3 - 3ax2 + 3a2x - a3 = (a - x)3
- Sum of Two Cubes: x3 + a3 = (x + a)(x2 - ax + a2)
- Difference of Two Cubes: x3 - a3 = (x - a)(x2 + ax +a2)
Sorry for the flood. :-)