Building Sentences: Dividing Polynomials

Dividing polynomials involves long division. We determine what we need to multiply the first term in the divisor by to get the first term in the dividend. Then multiply the entire divisor and subtract from the dividend. We line up powers of x in the quotient just like we lined up powers of 10 when we did long division of real numbers.

x+1

)

x3y+

x2y+

x2+

xy+

2x+

y+

1

Now I ask what do I need to multiply x by to get x³y?. The answer is x²y. I multiply x+1 by x²y and subtract.

			x2y
x+1	)	x3y+	x2y+	x2+	xy+	2x+	y+	1
		-x3y	-x2y

Now I bring down the rest of the dividend and ask what do I multiply x by to get x²? The answer is x. I multiply x+1 by x and subtract.

			x2y+	x
x+1	)	x3y+	x2y+	x2+	xy+	2x+	y+	1
		-x3y	-x2y
				x2+	xy+	2x+	y+	1
				-x2		-x

Now I bring down the rest of the dividend and ask what do I multiply x by to get xy? The answer is y. I multiply x+1 by y and subtract.

			x2y+	x+			y
x+1	)	x3y+	x2y+	x2+	xy+	2x+	y+	1
		-x3y	-x2y
				x2+	xy+	2x+	y+	1
				-x2		-x
					xy+	x+	y+	1
					-xy		-y

When I bring down what's left of the dividend I get x+1. If I multiply x+1 by 1 I get x+1. So I multiply x+1 by 1 and subtract. This time it comes out even.

			x2y+	x+			y+	1
x+1	)	x3y+	x2y+	x2+	xy+	2x+	y+	1
		-x3y	-x2y
				x2+	xy+	2x+	y+	1
				-x2		-x
					xy+	x+	y+	1
					-xy		-y
						x+		1
						-x		-1

So, (x³y+x²y+x²+xy+2x+y+1)÷(x+1)=x²y+x+y+1.

Building Sentences: Polynomials

A polynomial is an equation with a mix of numbers and variables. The terms of a polynomial are separate by operations. If a polynomial only has one term it is called a monomial. If it has two terms it is called a binomial. If it has three terms it is called a trinomial. All other polynomials are simply called a polynomial.

A polynomial has two parts. The real number part is called the coefficient and the letter parts are called the variable. In the polynomial 3x+2, x is a variable and 3 and 2 are the coefficients. Some of the parts of 3x+2 are not shown. The exponent on the x is 1. There is a variable that goes with the coefficient 2; remember from our discussion of exponents that anything raised to the 0 power is 1, so 2 is actually 2×1=2×xº. The terms where all of the variables are 0 powered are called constants.

The degree of a polynomial is the highest exponent of the polynomial. If x³ is the highest exponent in the polynomial we say it is a third degree polynomial. A polynomial of degree 1 is called a linear equation. A polynomial of degree 2 is called a quadratic equation. Degree 3 polynomials are called cubics. A 4th degree polynomial is called a quartic and a 5th degree polynomial is called a quintic. A linear equation draws a line when we graph it. All even degree polynomials graph a parabola. All odd degree polynomials approximate a line. I can't show you pictures in this blog, so we won't be able to cover graphing of polynomials. But a parabola looks like a u. All even degree polynomials enter a graph going one direction, then change directions at some point and leave the graph going in the opposite direction. All odd degree polynomials enter the graph in one direction and leave the graph in the same direction. Polynomials of degree higher than 2 sometimes bounce in the middle.

It is customary to write polynomials in descending degree order with coefficients first followed by variables in alphabetical order followed by radicals.

When adding two polynomials, you add only the terms that have the same variables with the same exponents. You simply add the coefficients.
(x²y³+xy)+(2x²y³+xy)=
((1)x²y³+(1)xy)+(2x²y³+(1)xy)=
(1+2)x²y³+(1+1)xy=
3x²y³+2xy.

If you're subtracting a polynomial from another polynomial, it's best to distribute the subtraction to all of the terms of the second polynomial at the start. You change the subtraction operation to an addition operation and change the sign of each of the terms of the second polynomial.
(2x²+3x+1)-(x²+x-1)=
(2x²+3x+1)+(-x²-x+1)
Now you drop the parenthesis and combine like terms.
2x²-x²+3x-x+1+1=
(2-1)x²+(3-1)x+(1+1)=
x²+2x+2

When multiplying two polynomials, we have to use the distributive property of real numbers. Every term in the first polynomial has to be distributed to each term of the second polynomial. Let's first multiply a polynomial by a constant.
3(x⁴+2x³+x²+3x+4)=
(3×x⁴)+(3×2x³)+(3×x²)+(3×3x)+(3×4)=
3x⁴+(3×2)x³+3x²+(3×3)x+12=
3x⁴+6x³+3x²+9x+12.

Next, let's multiply two binomials. We use distribution twice.
(x²y+2)×(xy²-1)=
x²y×(xy²-1)+2×(xy²-1)
Notice how the mutliplication sign and the second binomial get inserted into the first binomial. Now we distribute again.
(x²y×xy²)+(x²y×-1)+(2×xy²)+(2×-1)
Watch out for negative signs. Make sure they distribute with the number. Now, we multiply everything out. Remember to add exponents when the letters are the same.
x²⁺¹y¹⁺²-x²y+2xy²-2=
x³y³-x²y+2xy²-2

We'll leave dividing two polynomials until later.

The Properties of Real Numbers

Now that we have an alphabet, let's establish the rules of grammar for the language of mathematics. All of these rules work for any real numbers, so we'll use our new variables to express numbers in general. You can use the rules for substituting for variables with any of these rules, just be sure to use different numbers for different variables.

There are several kinds of rules. There are axioms which are so basic of a rule they are simply taken for granted. There are laws that are the consequence of axioms and are true everywhere all the time. And there are theorems which are rules that have been accepted as true because they have been built from axioms and laws and other theorems. Some of them we've already seen.

Closure Property
Adding any two real numbers results in another real number.
Multiplying any two real numbers results in another real number.
Transitive Property
If two numbers are equal to the same number, then they are equal to each other.	If a=c and b=c then a=b
Equality Axioms
If two numbers are equal and you add the same number to each the resulting numbers are equal.	If a=b then a+c=b+c.
If two numbers are equal and you mutliply each by the same number the resulting numbers are equal.	If a=b then ac=bc.
Commutative Property
When adding real numbers, the order doesn't matter.	a+b=b+a
When multiplying real numbers, the order doesn't matter.	ab=ba
Associative Property
When adding real numbers, the grouping doesn't matter.	a+(b+c)=(a+b)+c
When multiplying real numbers, the grouping doesn't matter.	a(bc)=(ab)c
Distributive Property
Multiplying a sum of real numbers by a real number is the same as multiplying each real number of the sum first then adding.	a(b+c)=ab+ac
This is a powerful property. It allows us to pull a common factor out of the parts of an equation and deal with it separately.
Identities
Adding zero to a real number doesn't change the number.	a+0=a
Multiplying a real number by 1 doesn't change the number.	a×1=a
Inverses
Adding the additive inverse of a real number to itself results in the identity.	a+(-a)=0=(-a)+a
Multiplying the reciprocal of a real number to itself results in the identity.	a(1/a)=1 if a≠0
Zero Principle
Multplying by zero results in zero.	a×0=0
If the product of two real numbers is zero, then one of them must have been zero.	If ab=0 then a=0 or b=0
Trichotomy Property
If two real numbers are compared, the second number can have only one of three outcomes, either greater than, less than or equal to.	a<b, a>b or a=b
Definition a^-1	a^-1=1/a
Properties of Negatives
Multiplying a real number by -1 changes the number's sign.	(-1)a=-a
	-(-a)=a
For any negative product of two real numbers, the negative sign can be assigned to either one the numbers.	(-a)b=-ab=a(-b)
The product of two negative real numbers is equal to the product of the additive inverses of the two numbers.	(-a)(-b)=ab
Negating the difference of two real numbers changes to signs of each number.	-(a-b)=b-a
Properties of Quotients
For any two rational numbers, the first rational number is equal to the second rational number if the product of the numerator of the first and the denominator of the second is equal to the product of the numerator of the second and the denominator of the first	a/b=c/d if and only if ad=bc
Mutliplying a rational number by a rational number equal to 1 changes the appearance of the rational number but not its value.	a/b=ad/bd
a/b+c/b=(a+c)/b
a/b+c/d=(ad+bc)/bd
a/b × c/d = ac/bd
a/b ÷ c/d = a/b × d/c = ad/bc
Definition
The principle square root of a number √a is the nonnegative real number b such that b squared is equal to a.	b²=a
Laws of Exponents
Definition
The absolute value of a real number is the magnitude of the number without a sign.	|a|=a, |-a|=a
Absolute Value Properties
The absolute value of a real number has the following properties
Positive Definite
For all real numbers, the absolute value of the number a is greater than or equal to 0 with |a|=0 only if a=0
Symmetric
For any two real numbers, |a-b|=|b-a|
Triangle Inequalities
For any two real numbers, the absolute value of the sum of the numbers is less than or equal to the sum of the absolute values of the numbers.	|a+b|≤|a|+|b|
For any two real numbers, the absolute value of the difference of the numbers is greater than or equal to the difference of the absolute values of the numbers.	|a-b|≥|a|-|b|
For any two real numbers, the absolute value of the difference of the absolute values of the numbers is less than or equal to the absolute value of the difference of the numbers.	||a|-|b||≤|a-b|

Variables

Mathematics often uses letters in place of numbers to illustrate a general principle. In a mathematical equation, the letters are referred to as variables because they can represent any number or numbers that can change depending on the situation. There are two types of variables; independent and dependent. An independent variable can change its value, usually without any restrictions. A dependent variable's value usually depends on the value of the independent variables in an equation. If a number alone is present in an equation, it represents a constant or a value that doesn't change.

Letters represent numbers sort of like letters can represent other letters in a code. If the letter A represents F in a code, the letter A represents F everywhere it appears. Similarly, if the letter a represents the number 5 in an equation, it represents 5 everywhere in the equation. The letter b would indicate another number besides 5.

Variables always have a combination of letters and numbers. Numbers in front of variables like 3a means there are three a's in the equation. If no number is in front of a variable it is understood that it is preceded by 1. Exponents are also always present with variables. If they're not explicitly shown they're undertood to be 1.

Variables can behave like numbers. Any operation you can perform on a number can be theoretically performed on a variable.

Multiplying a variable by itself changes the variables's dimension. If the variable a represents a line, then a² represents a square and a³ represents a cube and so forth. You can't add a, a² and a³ because they're not the same shape. You can only add lines to other lines, squares to other squares and cubes to other cubes. You can't add a² and b² because a² is a square and b² is a circle. To add variables together, they have to be the same letter and the same exponent. If the letters are the same and the exponents are the same, you add the numbers that come in front.

3a + 2a = 5a

When you multiply variables of the same kind together, you add the exponents. When dividing variables of the same kind, you subtract the exponent of the second variable from the first.

a²b × ab = a³b²

In a rational expression, a fraction involving variables and numbers, to add or subtract, the numbers and letters and exponents on the letters must be the same. If you have a² in one denominator and a³ in the other denominator, you multiply the first rational number by a/a. Remember that a fraction where the numerator and the denominator are the same is the same as 1, and multiplying a number by 1 doesn't change the value of the number. In this case, we only change the way the number looks.

1/a²b + 2/a³ =
(a/a × 1/a²b) + (b/b × 2/a³) =
a/a³b + 2b/a³b

Now that the denominators are the same we can add the two rational expressions. But notice that the numerators have different numbers and letters.

a/a³b + 2b/a³b = (a + 2b)/a³b

How would this work if we replaced the letters with numbers? Let's let a=3 and b=5.

1/3²5 + 2/3³ =
(3/3 × 1/3²5) + (5/5 × 2/3³) =
3/3³5 + 10/3³5 =
3/135 + 10/135 = 13/135

We can let a and b represent any two different numbers and we'll get a different result each time. Using the numbers only works for a specific case, but using variables allows us to talk about what happens in a general case. This is what variables are used for. We build mathematical sentences to describe what happens in a general sense using variables and then use specific numbers to explore a specific case. We can use variables to describe a general case of how long it takes to get to a destination at any speed, then put in numbers to see the results at specific speeds. To define how long it takes to get to a destination at any given speed we would write an equation using variables for distance and speed. If we let d represent the distance we want to travel and m represent the miles we travel in an hour, our equation would be d × 1/m.

Let's see what happens if we want to travel 100 miles at 55 mph. We put 100 in place of d and 55 in place of m.
100 × 1/55 = 1.812 hrs

To see what happens if we want to travel the same 100 miles at 65 mph we put 100 in place of d and 65 in place of m.
100 × 1/65 = 1.538 hrs

Using variables, we've set up a general case for how long it takes, then we can look at specific cases by substituting numbers for the variables. This is called modeling and is a very important topic in mathematics. All kinds of things can be modeled from the growth of tiny bacteria to landing a man on the moon. There is a famous equation developed by Isaac Newton to determine the gravity exerted by a body in space. All you have to do is substitute the size of the bodies and the distance between them. The same equation works everywhere in the universe, on Earth, on the Moon even on Jupiter or Mars. We have one equation and we can take it anywhere we want to go and substitute numbers for the variables depending on where we are.

Building Simple Words: Exponents and Roots

Multiplication is multiple additions. Exponentiation is multiple multiplications. Exponents are written as superscript numbers above and to the right of the number to be exponentiated. 2×2×2 is written as 2³ and equals 8. This is read as 2 raised to the 3rd power. Not all exponents can be shown in this format, so I will use a carat (^) to indicate the number that follows is the exponent.

Numbers raised to the second power are said to be squared because the area of a square is the product of the two equal dimensions (length, width) multiplied together. Numbers raised to the third power are said to be cubed because the volume of a cube is the product of the three equal dimensions (length, width, height) multiplied together. All other exponents use the ordinal form of the exponent (to the fourth, to the fifth, to the sixth, ect.) Any number raised to the 0 power equals 1.

Squares are so common that the squares from 1 to 10 should be memorized. They are taken from the multiplication table.

1² = 1×1 = 1
2² = 2×2 = 4
3² = 3×3 = 9
4² = 4×4 = 16
5² = 5×5 = 25
6² = 6×6 = 36
7² = 7×7 = 49
8² = 8×8 = 64
9² = 9×9 = 81
10² = 10×10 = 100

You can't directly add numbers with exponents. Each must be exponentiated first, then the results are added together. With multiplication, if the numbers are the same the exponents are added together. 2^3×2^4=2^(3+4)=2^7. If you exponentiate an exponent, you multiply the exponents. (2^3)^4=2^(3×4)=2^12. With a rational number, when the number in the numerator and the number in the denominator are the same, you subtract the exponents. 2^4/2^3=2^(4-3)=2^1=2. If the exponent in the numerator is larger, you subtract the exponent in the denominator from the exponent in the numerator and replace the denominator with 1. 2^6/2^3=2^(6-3)/1=2^3/1=2^3. If the exponent in the denominator is larger, you subtract the exponent in the numerator from the exponent in the denominator and replace the numerator with 1. 2^3/2^6=1/2^(6-3)=1/2^3.

Just as division is the opposite of multiplication, there is an opposite operation for exponentiation. The root of a number is the number that when multiplied by itself gives the orignal number. Roots are indicated with a symbol called a surd √ with an index over the little bar. The index shows the root and the surd separates the number to be broken down. The index and surd together with a number are called a radical. If the root is the square root the index is not shown. ³√27 = 3 because 3³=27.

Like with addition of exponents, you can't add different radicals. You have to get the roots first, then add the resulting numbers. We can mix numbers and radicals. A number can be added to a radical or a radical can be multiplied by a number. 3+√5 is a valid number as is 3√5. We can add two radicals together if they are exactly the same. In that case, you add the numbers on the outside and keep the radical unchanged. 4√5 + 3√5 = 7√5. I can't add 3√25 and 4√36 because the radicals are different. I would have to convert the radicals to numbers then add the result. 3√25+4√36=3(5)+4(6)=15+24=39.

Roots can be expressed as rational exponents. A square root is a ½ exponent. √4=4^½. If the number inside the radical is raised to a power, the power inside the radical becomes the numerator of the rational exponent and the index of the radical becomes the denominator of the rational exponent. Converting roots to exponents means we can use our rules for multiplying and dividing exponents to multiply two roots together. If the numbers inside the radical are the same we simply add the exponents together using the rules for adding and subtracting rational numbers. If the exponent on the number under to surd is the same as the index of the root, the index and the exponent cancel each other out and the surd disappears. ³√27³=27^(3/3)=27^1=27. If the indexes are the same, multiply the numbers under the surd and keep the index. ³√9׳√8=³√(9×8)=³√72.

To simplify a radical you remove groups of common factors based on the index. First you factor the number under the surd into its prime factors. Then you remove groups of factors based on the index. When all of the groups have been removed, you multiply the numbers on the outside together and multiply the numbers remaining under the radical. Let's simplify ³√72. The prime factors of 72 are 3×3×2×2×2. The index is 3 so we want to remove factos in groups of 3. There are 3 2's so we remove them from inside the surd and place a single 2 on the outside;2³√(3×3). There are only 2 3's under the surd and we need to pull out in groups of three, so the 3's stay under the surd. Multiply the 3's back together again and ³√72=2 ³√9. How about ³√1296. First we factor 1296 into its prime factors; ³√(3×3×3×3×2×2×2×2). We need to remove factors in groups of 3. There are 4 3's, so we can remove 3 of them and there are 4 2's so we can remove 3 of them as well; (3)(2)³√(3×2). There are no more groups of 3 left so we multiply the numbers on the outside and multiply the numbers on the inside; 6 ³√6.

Roots of prime numbers don't have an even representation because there are no factors of prime numbers, so you can't get a prime number by multiplying any number to itself. All roots of prime numbers are irrational numbers. We'll discuss them next.

Even roots of negative numbers are also a problem. If you think about the rules for multiplying negative numbers, you can see where the problem lies. Remember that if there are an even number of negative numbers being multiplied together, you get a positive number. If there are an odd number of negative numbers, you get a negative number. A negative number exponentiated an even number of times gives a positive number. So there is no way to get a negative number if the exponent is even, thus there can't be an even root of a negative number. Finding an even root of a negative number requires us to use a very special number called the imaginary number. We won't worry about the imaginary number for now.

Modern calculators usually have a key for finding any root of any number, but for a method for finding any root of any number see this article in Wikipedia Nth Root Algorithm. Now that we know about roots, let's talk about the last group of numbers that make up our alphabet.

The Alphabet of Math: The irrational numbers

The last group of numbers that we are going to discuss are irrational numbers. Remember that rational numbers are numbers that can be represented as a ration of two numbers. Irrational numbers are numbers that can't be represented as a ration of two numbers. Irrational numbers fill in the spaces between rational numbers on the number line.

The Greek Pythagorean Society studied numbers and their effect in the universe. They believed all numbers were rational. When one of them asked what the square root of 2 was, they were so upset by the resulting irrational number that they took out in a boat, tied him up and threw him into the sea! Mathematicians are more receptive to new ideas these days.

Rational numbers are numbers that are either terminating or repeating and nonterminating. Irrational numbers are nonrepeating and nonterminating. Many numbers fall in this category. The roots of prime numbers are all nonrepeating and nonterminating. Roots of prime numbers are all irrational. An ancient special number is irrational. Pi is the ratio of the diameter of a circle to its circumference. This ratio has been known for thousands of years, but everyone in antiquity used a rational approximation to pi. Today, we recognize that pi can't be represented as a a rational number so we simply use the Greek letter π to represent it. Another special number is used to find an exponential growth like with rabbits or bacteria. It's called e and it's irrational also.

With the natural numbers, the integers, the rational numbers and the irrational numbers our alphabet is essentially complete. These numbers together are called the real numbers.

With our alphabet complete, we can begin talking about all topics in mathematics.

The Alphabet of Math: Decimal numbers

Rational numbers are actually the division of whole numbers into smaller parts. A fraction represents a division problem. The numerator is the dividend and the denominator is the divisor. So 7/4 could also be written as 7÷4. 4 divides 7 once with a remainder of 3. We can represent this as a mixed number, a whole number for the whole number of divisions and a rational number for the remaining parts of the whole left over. So 7/4 is the same as 1 ¾. But, we won't represent rational numbers with mixed numbers. We are going to use a new type of number that looks more like whole number called a decimal number.

A decimal number has two parts, a whole number part and a fractional part separated by a decimal. To get a decimal representation of a rational number we use long division and instead of stopping when we get to the remainder, we keep dividing until we get to a zero remainder. 7/4 would be divided like this

		1
4	(	7
	-	4
		3

This is where we have been stopping. But we can continue the division process to get a decimal representation by placing a decimal after the 1 and using zeros to continue dividing until we get a zero remainder.

		1.	7	5
4	(	7
	-	4
		3	0
	-	2	8
			2	0
		-	2	0
				0

So 7/4 = 1.75

This is a special decimal number called a non-repeating, terminating decimal. This decimal number doesn;t repeat and it comes out to a zero remainder. What happens if the number doesn't come out to a zero remainder?

		2.	3	3
3	(	7
	-	6
		1	0
	-		9
			1	0
		-		9
				1

This decimal number repeats 3 over and over again. This is called a repeating decimal number. We usually only show the repating part once and draw a bar opver the top of the repeating part.

All rational numbers can be represented as either non-repeating, terminating decimals or repeating decimals.

Adding decimal numbers is done just like whole numbers. Line up the decimals then add the numbers just like whole numbers.

	1	.	5	4	3
+	2	.	3	4
	3	.	8	8	3

Multipying decimal numbers is similar to multiplying whole numbers. Multiply the numbers just like whole numbers disregarding the positions of the decimals. After you've finished multiplying, count the number of digits to the right of the decimal in each number. This is the total number of decimals that will be in the product. Starting from the right, you count positions to the left for as many decimals as you have in the multiplicand and multiplier and place the deicmal in that position.

	1	.	5	4	3
×		2	.	3	4
		6	1	7	2
	4	6	2	9
3	0	8	6
3	6	1	0	6	2

There are 3 decimals in the multiplicand and 2 in the multiplier so our answer needs 5 decimals. Starting from the right we count back 5 positions and place a decimal between 3 and 6. So 1.543×2.34=3.61062

Dividing decimals works just like dividing we have seen above but we start by making the divisor into a whole number by miving the decimal to the right until there are no more numbers. We also have to move the deicmal in the dividend the same number of places to the right. The decimal in the quotient will be at the same position as the new position in the dividend.

2.

3

4

(

3.

6

1

0

6

2

There are 2 decimal positions in the divisor. So move the decimal 2 places to the right in the divisor and the dividend. Now divide as usual.

						1.	5	4	3
2	3	4	(	3	6	1.	0	6	2
			-	2	3	4
				1	2	7	0
			-	1	1	7	0
					1	0	0	6
					-	9	3	6
							7	0	2
						-	7	0	2
									0

What happens if the quotient doesn't repeat and doesn't terminate? This is the last of our number types for our alphabet. They are called irrational numbers because they can't be represented as rational numbers. There are lots of irrational numbers and we'll discuss them in another post.