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# Polynomials n' Stuff

math is a wonderful thing math is a really cool thing
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on 25 April 2011

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#### Transcript of Polynomials n' Stuff

POLYNOMIAL A polynomial is an expression made of one or more variables and constants, using only +
and -, with positive whole exponents. 2x + 5x + 6 7x + 6 (x+5) (x-3) 2 3 A DOUBLE ROOT is a part of a polynomial that is squared. A TRIPLE ROOT is a part of a polynomial that is cubed.
In a graph, a double root will behave like so: (GRAPH HERE) In a graph, a TRIPLE ROOT will behave like so: (graph here) 7x +8x -2x +5 3 2 In a polynomial, the DEGREE is the value of the HIGHEST EXPONENT. The degree of the above polynomial would, for instance, be 3. 3y + 2 3y + 2 COMBINING LIKE TERMS This expression has four parts, or rather, TERMS. Out of these four terms, both 2x and 5x are alike, and can be combined to make 7x, and make the expression have 3 terms instead of four. A MONOMIAL
has one term. examples: 8, 4x, y A BINOMIAL has two terms. examples:
8x + 3,
4y -5,
7 + X 2 (Lastly, a TRINOMIAL has three terms.
I believe that you get the picture by now.) You could also say this by stating the root has a MULTIPLICITY of 2 or 3. In STANDARD FORM, the polynomial is written in order of decreasing exponential value. example: 2x + 3xy - 5x +x -6 4 3 2 In a polynomial's factored form, however, it is easier to find the ROOTS, also known as the X-intercepts. (x+5)(x-3) The ROOTS of this polynomial would be NEGATIVE 5 and POSITIVE 3. The graph of the polynomial would pass through both of these. Sometimes, the graphs of the polynomials end in different ways. If the right arrow/right end of the graph is pointing DOWN, then the polynomial is negative. You can also determine this from the equation, by seeing if there is a negative distributed to it out in front. If the right arrow is pointing UP, then the polynomial is positive. There is no negative distributed in the front of the equation. You can refer to the right arrow's direction as the polynomial's END BEHAVIOR. You can do various operations with polynomials, including addition, subtraction, multiplication, and (long) division. When adding polynomials, simply combine all like terms. (5x + 2y) + (3x + 4y + 6) 8x + 6y + 6 Do the same for subtraction. (8x + 6 ) - (4x + 5) = 4x - 1 When multiplying polynomials, you have to "foil", or rather, distribute very term to every other term. (x + 5x)(3x + 2x + 4) 2 2 3x + 17x + 14x + 20x 4 3 2 Videos explaining these concepts from Khan Academy POLYNOMIAL LONG DIVISION The process of dividing polynomials may seem arduous and complicated, but it actually parallells exactly with the long division of constants without variables or exponnents. STEP 1: set up problem x + 5 x + 2x -8x + 35 4 3 STEP 2: divide first term of divisor into first term of divedend x + 5 x + 2x -8x + 35 4 3 A PLACEHOLDER is a term that is needed when exponents skip a degree in the dividend. All placeholders are 0x, and they will to be to whatever degree is missing. the negative 300 we get as a remainder can be written as an addition to the quotient, as a fraction, with -300 on top and x+5 on the bottom. -300 x+5 sum difference product AND NOW FOR SOMETHING COMPLETELY DIFFERENT. i is an imaginary number. IMAGINARY NUMBERS are the result of trying to take the square root of a negative. for example, i is the square root of negative 1. i behaves like a variable, but can at times, be turned into a real number. Since i is the square root of a negative one, i squared can be substituted for a negative one. Thus, in situations where substitution is not possible, i behaves like a regular variable, and can attatch itself to constants and be used in equations and operations.
For example, take 4 +3i. Whenever a quantity of i (here it's 3) and a real number (4) are combined in an expression, this is called a COMPLEX NUMBER. OPERATIONS WITH i AND THE COMPLEX NUMBER To multiply i with a real number, treat i like a variable. EXAMPLE: 3(5i) = 15i To multiply two complex numbers, it's the same as multiplying two normal expressions, except for one exception. Whenever i occurs, replace it with a negative one. 2 EXAMPLEs: (2i)(4i)= 8i = -8
(3+4i)(6-2i)= 18-6i-24i-8i = 10-30i 2 2 When dividing with complex numbers, you'll need to use the COMPLEX CONJUGATE in the denominator, which is a complex number that will cancel another out. [ex: 2-6i and 2+6i, 3-4i and 3+4i.] So, let's take 2/3-i. To divide by 3-i, we multiply the whole equation by the complex conjugate, 3+i/3+i. On the top, you'll have 6+2i, and on the bottom, you'll get 9-i , which is the same as 10. 2 This can further simplify out to 6/10 + 2i/10, or rather, 3/5 + i/5. To add and subtract complex numbers, treat i like a normal variable, and combine like terms. [EX. (5-2i) + (7+3i) = 12+i] QUADRATICS AND
IMAGINARY/COMPLEX
NUMBERS Quadratics, and the graphs of them, (parabolas) can also relate directly to imaginary/complex numbers. Most of the time, a parabola has two x-intercepts, or two roots. These are both real, rational numbers. However, sometimes the parabola floats in the middle of the plane, and does not run into the x-axis. In cases like this, the roots are considesred complex, and contain an imaginary/complex numbe
r. To find these complex roots, you have to input the quadratic into the quadratic formula. Once the numbers are in the quadratic formula, you have to look at the part under the sqrare root symbol on top. (the square root of b squared minus 4ac) If the value under the square root is negative, the quadratic will have complex solutions. If it is positive, it will have 2 normal, real, x-intercepts/roots. For example, the equation
y=2x + 5x-3 will equate out to {square root of} 25-(4)(2)(-3), which is {square root of} 25+24, positive 49, which is 7. ON the other hand, take the equation y= 2x +5x+4. UNder the square root symbol, it comes out to 25-32, which is negative, and thus the roots are complex. 2 2 Now, since i is equal to negative one, the other powers of i also have different values. But they are easy to remember, since they follow a pattern. 2 i is negative i, i is equal to POSITIVE one, i is simply equal to i, and i is equal to negative one again. 3 4 5 6 So, how would we figure out something like i or i ? 17 23 First, we know i is equal to one. 16 is divisible by 4, and there's one left over to get to 17. So, we go up the pattern one number (from i , 1, to just plain i. As for i , there are 3 left over after being divided by 4 (i ) so we go up 3 in the pattern to -i. 4 4 23 20 Different forms of quadratics/polynomials The normal form of any quadratic/polynomial is standard form, y= ax +bx + c. 2 To get to factored form, which shows the x-intercepts, or roots, of the equation, simply factor it completely. For example, x +2x -8 would factor out to (x+4)(x-2), and the roots would be POSITIVE two and NEGATIVE four. You reach this by using "difference of squares", a formula that makes facroting a bit easier. The trick is: the numbers you input need to add up to the b in the formula, and multiply out to get c. (+4 and -2 added is 2, and multiplied, they're -8. 2 Vertex form does exactly what is sounds like; it shows the vertex.
The form for this is y= a(x-h) +k. 2 To get to this form fromstandard form, you have to complete the square. COMPLETING THE SQUARE In order to "complete the square", you start with an equation in standard form. Let's use x -2x+15.
First, we set it equal to 0. Then, we subtract 15 and put it onthe other side of the equals sign, but we leave a blank where it used to be, so we have x -2x+ ___ = -15. 2 2 Next, we find a number to put in the blank space. To get this number, we take half of the b, and square it. In this case, we have 2. Half of 2 is one, 1 squared is still one. (If instead we had something like 6x before the blank, we would take half of 6, which is 3, and square it to get 9. Anyway, moving along.) Since we added 1 to one side of the equation, we have to add it to the other as well. -15 plus one is -14. Then, with the other half of the equation, we put in factored form. [x -2x+1 becomes (x-1) ] 2 2 Lastly, we bring the 14 back to the other side. So, we are left with the equation in vertex form: (x-1) +14=y. 2 The vertex here would be 1, 14. Transforming functions/ quadratics Also, i can be graphed, but not on a normal x-y plane. Imaginary numbers get their own plane, the COMPLEX PLANE. It looks the same as a standard graph, except the "y-axis" is now the imaginary axis, and the "x-axis" is the real axis.
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