Algebra
Children categories
For all real numbers A, B and for when C does NOT equal 0, the equations:
A = B and AC = BC are equivalent to one another.
Each side can be multiplied by the same NONZERO number without changing the solution set.
Definition:
A polynomial function of a real variable \(x\), is a function defined by the sum:
\[ P(x) = \sum_{k=0}^{n}c_k x^{k} = c_0 + c_1 x + c_2 x^{2}... + c_n x^{n}. \]
Sets, Rings, and Groups of Numbers
A linear function, \(f(x)\), of a real number, \(x\), is defined by two properties:
- for any real number, \(a\), \(f(ax) = a(f(x));\)
- for any real number, \(h\), \(f(x+h) = f(x) + f(h).\)
For all real numbers A, B and C, the equations:
A = B and A + C = B + C are equivalent to one another.
The same number may be added to each side of an equation without altering the solution set.
The product of two linear factors
\[ (x - \alpha)(x - \beta)\]
is a polynomial of degree two:
\[(x - \alpha)(x - \beta) = x^{2} - (\alpha + \beta)x + \alpha\beta.\]
Definition:
Polynomials of degree two are also called quadratic polynomials.
Very simply stated, the Distributive Property is easy to understand when seen like this:
a(b+c) = ab + ac
OR
(b+c)a = ba + ca
In more complex examples, the Distributive Property can be applied to equations like this one, though the same principles still apply:
EXAMPLES:
3(x + y)
= 3x + 3y
-2(5 + k)
= -2(5) + (-2)(k)
= -10 - 2k
4x + 8x
= (4 + 8)x
= 12x
Definition:
The roots of a polynomial \(P\) are the solutions of the equation:
\[ P(x) = 0. \]
For any real number \( a \neq 0 :\)
\( a + (-a) = 0 \ \) and,
\( a \left( \frac{1}{a} \right) = 1.\)
That is, any number times its reciprocal equals 1.
The reciprocal of zero is not defined.
How do we divide polynomials?
Suppose, for example that you are given the problems:
Express \(\frac{x^{2} - 1 } {x - 1 } \) in terms of \(x\).
Find \(\frac{x^{3} + 2x^{2} + x}{x + 1}\) in terms of \(x.\)
What do we do?
For any real number a,
a + 0 = 0 + a = a
a * 1 = 1 * a = a
Notes: An easy way to remember this is that the Identity Property leaves the IDENTITY of a real number unchanged. Adding 0 to any number or multiplying any number times a does not change the value of the number in any way.
EXAMPLES:
12m + m
= 12m + 1m
= (12 + 1)m
= 13m
The Commutative Property states:
For any real numbers a and b:
a + b = b + c
ab = ba
The Commutative Property is when the ORDER CHANGES but the result remains the same.
The Associative Property states that for any real numbers a, b and c:
a + (b + c) = (a + b) + c
a(bc) = (ab)c
With the Associative Property, parentheses amongst 3 terms/factors change, but the order of the terms stays the same.
All equations have various parts. The variable represents the unknown. We can tell what "x" represents in this particular equation, almost by guessing. WHAT + 23 = 45?
The VARIABLE here is "x"
The CONTANT is 23, because it is "constantly" there, and we have already identified its value as 23.
Mathematically, we can obtain what "WHAT" equals, by subtracting 23 from 45.
45 - 23 = 22
Therefore, x = 22
Definition:
For any natural number \(n\), and any natural number \(i\), such that, \( n \geq i \geq 0\):
The binomial coefficient,
\[ \binom{n}{i}, \]
of \(n\) and \(i\) is,
\[ \binom {n}{i} := \frac{n!}{(n-i)!(i)!}. \]
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A negative number is any number to the left of 0 on the number line.
By contrast, a positive number is any number to the right of 0 on the number line.
For a function \(f(n)\) of a natural number \(n,\) we can prove that \(f(n)\) has the property P, for all \(n\), if we can do two things:
- prove that P is true for \(f(n)\) when \(n\) = 1;
- second: prove that if P is true for \(f(n)\), then P is also true for \(f(n+1)\).
This process is called mathematical induction.
We define \(n\) factorial, written \(n!\), by induction.
For \(n = 1,\)
\[ n! = 1! = 1. \]
For \((n+1)\),
\[ (n+1)! = (n+1)(n!). \]
The technical rule to remember here is that for any real number a:
(-1)(-a) = a.
For a finite set \(S\) with \(n\) elements, the total number of subsets of \(S\) with \(k\) elements
\[ ( 0 \leq k \leq n) \]
is
\[ \binom{n}{k}.\]
Absolute value is the distance on the real number line between any given number and zero.
Absolute value can never be negative, because technically distance can NEVER be negative.
Distance is always positive.
Absolute value is denoted between | number | - so to the reader, it will appear as: |-3| = 3.
Absolute value is the distance on a number line between any given point and zero.
Therefore, because distance is always POSITIVE (you cannot have a negative distance), absolute value is also always POSITIVE.
For example, if you have |x| = 2
This means that x = -2 or 2
WHY? Because distance is always POSITIVE.
Therefore, remember: Any variable in brackets could have TWO values, one positive and one negative.
A set is a collection of "elements," or "members," and each set is entirely determined by its members. If \[ x \]is a member of a set \[ U, \] we write \[ x \in U. \] In basic algebra, the elements of a set are usually numbers.
We can also designate a set by enclosing its members in braces , { }.
Sets of Numbers Examples Sets: Natural Numbers, Whole numbers, Integers
By MandyCounting Numbers:
{1, 2, 3, 4, 5, 6, ...}
BY THE WAY, "..." notates continuation going up infinitely.
The set of Natural Numbers \( \mathbb{N}\ \ \) = {0, 1, 2 3, 4, 5, 6...}.
The set of Integers, \( \mathbb{Z}\ \ \) = {..., -3, -2, -1, 0, 1, 2, 3}.
The set of Rational numbers, \( \mathbb{Q}\ \ \) is the set of ratios of integers.
That is, every \(q \in \mathbb{Q}\ \)
is of the form:
\( q = \frac{n}{m} \ \)
With \( n, m \in \mathbb{Z}\ \) .
The set of Real Numbers \( \mathbb{R}\ \ \) is the set of limits of sequences of rational numbers.
Equations and Expressions are closely related.
The primary difference between the two is an equals sign. An "equation" has a left side, a right side and an equals sign separating the sides. An "expression," by contrast, doesn't have any "sides" and is simply what the name suggests: An algebraic "expression." Though sometimes it is possible to combine like terms, we are generally not expected to "do" or "solve" anything regarding expressions.
For example:
3x - 7 = 2
This is an EQUATION, because it has a left side, a right side, and an = sign separating the two.
3x - 7
This is an EXPRESSION, because there are no "sides" and no = sign.
As the name suggests, a linear equation in one variable implies that there is only ONE variable, and that the equation involves only real numbers. A linear equation in one variable can be written in this form: Ax + B = C where A does NOT equal zero.
A linear equation is also a first-degree equation, since the greatest power of any variable is 1.
Here are some examples of linear equations in one variable:
x + 2 = -1
x - 3 = 5
3k + 4 = 10
Here is a list of steps to remember when solving Linear Equations in One Variable.
You may want to become familiar with these steps, each one on its own before putting them all together!
STEP 1: Clear fractions. Get rid of any fractions you see in the original equation by multiplying each side by the least common denominator.
STEP 2: Simplify each side. Use the distributive property to get rid of parentheses and/or combine like terms as necessary.
STEP 3: Isolate the variable terms (often "x") on one side. Use varying properties to get all terms with variables on one side of the equation, with all nubers on the other.
STEP 4: Isolate the variable. Use properties to get an equation with just the variable on one side.
STEP 5: CHECK your solution. Substitute your answer back into the ORIGINAL equation to make sure it is correct.
4x - 2x - 5 = 4 + 6x + 3 |
Goal is to isolate x on one side to be able to solve. |
2x - 5 = 7 + 6x |
Combine like terms. |
2x - 5 + 5 = 7 + 6x + 5 |
Add 5 to each side. |
2x = 12 + 6x |
Subtract 6x from each side. |
-4x = 12 |
Then, divide both sides by -4 |
x = -3 |
CHECK your solution by plugging back into the original equation. |
Disclaimer: I did not create nor do I own these videos. I have simply embedded them, courtesy of YouTube. (But I do think this teacher does a fantastic job with his video tutorial series.)
Step-By-Step How To Solve A Linear Equation Using The Distributive Property
By Mandy
2(k - 5) + 3k = k + 6 |
Use the distributive property to simplify and combine like terms. |
2k - 10 + 3k = k + 6 |
Combine like terms. |
5k - 10 = k + 6 |
Then, add 10 to both sides. |
5k = 16 + k |
Then, subtract k from both sides. |
4k = 16 |
Then, divide both sides by 4 |
k = 4 |
CHECK your solution by plugging back into the original equation. |
Disclaimer: I did not create nor do I own these videos. I have simply embedded them, courtesy of YouTube. (But I do think this teacher does a fantastic job with his video tutorial series.)
The official formula is: D = R * T
Distance = Rate x Time
This can also be flipped around to be expressed as R = D / T (Rate = Distance / Time)
Or it can be flipped around yet again to be expressed as T = D / R (Time = Distance / Rate)
This formula is completely liquid and can be "remelded" as needed depending on what you're working on.
We can view the expression:
\[ \frac{X}{Y} \]
as asking, and answering, the question:
How many \(Y\)s are there in each \(X\)?
So for example if \(X = 1 \) and \(Y = 2,\)
then the fraction:
\[ \frac{X}{Y} = \frac{1}{2} ,\]
is asking:
How many twos are there in one?
And the answer is one-half.
Given two fraction, say, \(\frac{1}{3},\) and \(\frac{4}{5},\) we can multiply them together by multiplying the two numerators (the numbers upstairs) and the two denominators (the numbers downstairs), separately.
So for example:
\[ \frac{1}{3} \times \frac{4}{5} = \frac{ ( 1 \times 4 ) }{( 3 \times 5)} = \frac{ 4}{15}.\]
Or in general:
\[ \frac{P}{M} \times \frac{ Q}{N} = \frac{ P \times Q }{ M \times N}.\]
When we multiply fractions we are really taking fractions of fractions. Imagine that we have divided a pie into five equal parts. One slice falls on the floor and so we have four slices left; i.e., we have \(\frac{4}{5}\) of the original pie. Now even though we had cut out pie into five pieces, because we had expected five people to show up for our party, only three people actually show up.
We want to divide what is left of the pie up evenly among our three guests, and so each guest gets
\[ \frac{1}{3} \times \frac{4}{5} = \frac{ ( 1 \times 4 ) }{( 3 \times 5)} = \frac{ 4}{15}\]
of a pie.
In practice, this means that we would take each of the pieces we originally cut, and cut them all into three parts. Then we would give each guest four of these smaller pieces.
Because we had originally cut the pie into five pieces, and then cut each one of these pieces (except the one that fell on the floor) into three pieces, each little piece is equal to one fifteenth of a pie. Then we give each guest for each of these smaller pieces.
\[ \frac{ 4}{15}.\]
Suppose we have two pies.
One pie is cut into three equal pieces and the second one is cut into five equal pieces.
Now suppose we want to add a piece from the first pie to a piece from the second pie. How much pie do we have?
As an equations this reads:
\[ \frac{1}{3} + \frac{1}{5} = ? \]
As a mathematical expression, there is nothing wrong with:
\[ \frac{1}{3} + \frac{1}{5}\]
but intuitively we don't really know how much pie this is.
We need to be able to compare \(\frac{1}{3}\) to \(\frac{1}{5}\) and to be able to add these two quantities together.
What we need is a common denominator.
What is Percent?
A percent is a special ratio or fraction that always has denominator equal to one hundred.
For example,
\[ \frac{1}{100},\]
\[ \frac{10}{100},\]
\[ \frac{50}{100},\]
are all percents, called one percent, ten percent, and fifty percent, respectively.
Changes in percent can be counter-intuitive.
For example, if the value of a stock falls by fifty percent on one day, and then rises by fifty percent on the next, it will not have recovered all of its value.
It will in fact be seventy-five percent of its original value.
It would need to increase by one hundred percent to recover all its value.