home

=Welcome to Algebra in class=

FOR THE PURPOSE OF THE TEACHING WE WILL BE FOCUSING ON THE THREE COMMONLY KNOWN ALGEBRA EQUATIONS. THE THREE EQUATION THAT WILL BE DISCUSSED ARE:
 * 1) QUADRATIC EQUATION
 * 2) FACTORASING
 * 3) COMPLETING THE SQUARE

What is an Equation
An equation says that two things are equal. It will have an equals sign "=" like this: That equations says: **what is on the left (x + 2) is equal to what is on the right (6)** So an equation is like a **statement** "//this// equals //that//"
 * **//x//** || **+** || **2** || **=** || **6** ||

Parts of an Equation
So people can talk about equations, there are **names** for different parts (better than saying "that thingy there"!) Here we have an equation that says 4x-7 equals 5, and all its parts: A number on its own is called a **Constant**. A **Coefficient** is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient) An **Operator** is a symbol (such as +, ×, etc) that represents an operation (ie you want to do something with the values). || An **Expression** is a group of terms (the terms are separated by + or - signs) || So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"
 * [[image:http://www.mathsisfun.com/algebra/images/variable-constant.gif width="300" height="137"]] || A **Variable** is a symbol for a number we don't know yet. It is usually a letter like x or y.
 * [[image:http://www.mathsisfun.com/algebra/images/expression-term.gif width="202" height="133"]] || A **Term** is either a single number or a variable, or numbers and variables multiplied together.
 * [[image:http://www.mathsisfun.com/algebra/images/expression-term.gif width="202" height="133"]] || A **Term** is either a single number or a variable, or numbers and variables multiplied together.

Exponents
Examples: Exponents make it easier to write and use many multiplications Example: **y4z2** is easier than **y × y × y × y × z × z**, or even **yyyyzz**
 * [[image:http://www.mathsisfun.com/images/8-squared.gif width="181" height="160" caption="8 to the Power 2"]] || The [|exponent] (such as the 2 in x2) says **how many times** to use the value in a multiplication.
 * 82 = 8 × 8 = 64**
 * y3 = y × y × y**
 * y2z = y × y × z** ||

Polynomial
Example of a Polynomial: **3x2 + x - 2** A [|polynomial] can have **constants**, **variables** and the **exponents 0,1,2,3,...** And they can be combined using addition, subtraction and multiplication, ... **but not division!**

Monomial, Binomial, Trinomial
There are special names for polynomials with 1, 2 or 3 terms:

Like Terms
[|Like Terms] are **terms** whose variables (and their [|exponents] such as the 2 in x2) are the same. In other words, terms that are "like" each other. (Note: the **coefficients** can be different)

Example:
Are all **like terms** because the variables are all **xy2.** For further infomation tou can go to __http:www.mathisfun.com/exponents.html__ =**__2.Completing the square__**= To view how to complete the square go to __[|www.wikipedia/w/index/php?title=completing] the square and action.__
 * (1/3)xy2 || -2xy2 || 6xy2 ||

Background
There is a simple formula in [|elementary algebra] for computing the [|square] of a [|binomial]: For example: In any perfect square, the number //p// is always half the [|coefficient] of //x//, and the [|constant term] is equal to //p//2.

Basic example
Consider the following quadratic [|polynomial]: This quadratic is not a perfect square, since 28 is not the square of 5: However, it is possible to write the original quadratic as the sum of this square and a constant: This is called **completing the square**.

General description
Given any [|monic] quadratic it is possible to form a square that has the same first two terms: This square differs from the original quadratic only in the value of the constant term. Therefore, we can write where //k// is a constant. This operation is known as **completing the square**. For example:



Non-monic case
Given a quadratic polynomial of the form it is possible to factor out the coefficient //a//, and then complete the square for the resulting [|monic polynomial]. Example: This allows us to write any quadratic polynomial in the form

Formula
The result of completing the square may be written as a formula. For the general case:[|[][|1][|]] Specifically, when //a=1//:

Relation to the graph
Graphs of quadratic functions shifted to the right by //h// = 0, 5, 10, and 15. Graphs of quadratic functions shifted upward by //k// = 0, 5, 10, and 15. Graphs of quadratic functions shifted upward and to the right by 0, 5, 10, and 15. In [|analytic geometry], the graph of any [|quadratic function] is a [|parabola] in the //xy//-plane. Given a quadratic polynomial of the form the numbers //h// and //k// may be interpreted as the [|Cartesian coordinates] of the vertex of the parabola. That is, //h// is the //x//-coordinate of the axis of symmetry, and //k// is the [|minimum value] (or maximum value, if //a// < 0) of the quadratic function. In other words, the graph of the function //ƒ//(//x//) = //x//2 is a parabola whose vertex is at the origin (0, 0). Therefore, the graph of the function //ƒ//(//x// − //h//) = (//x// − //h//)2 is a parabola shifted to the right by //h// whose vertex is at (//h//, 0), as shown in the top figure. In contrast, the graph of the function //ƒ//(//x//) + //k// = //x//2 + //k// is a parabola shifted upward by //k// whose vertex is at (0, //k//), as shown in the center figure. Combining both horizontal and vertical shifts yields //ƒ//(//x// − //h//) + //k// = (//x// − //h//)2 + //k// is a parabola shifted to the right by //h// and upward by //k// whose vertex is at (//h//, //k//), as shown in the bottom figure.

Solving quadratic equations
Completing the square may be used to solve any [|quadratic equation]. For example: The first step is to complete the square: Next we solve for the squared term: Then either and therefore This can be applied to any quadratic equation. When the //x//2 has a coefficient other than 1, the first step is to divide out the equation by this coefficient: for an example see the non-monic case below.

Irrational and complex roots
Unlike methods involving [|factoring] the equation, which is only reliable if the roots are [|rational], completing the square will find the roots of a quadratic equation even when those roots are [|irrational] or [|complex]. For example, consider the equation Completing the square gives so Then either so In terser language: Equations with complex roots can be handled in the same way. For example:

Non-monic case
For an equation involving a non-monic quadratic, the first step to solving them is to divide through by the coefficient of //x//2. For example:

Integration
Completing the square may be used to evaluate any integral of the form using the basic integrals For example, consider the integral Completing the square in the denominator gives: This can now be evaluated by using the [|substitution] //u// = //x// + 3, which yields

Complex numbers
Consider the expression where //z// and //b// are [|complex numbers], //z//* and //b//* are the [|complex conjugates] of //z// and //b//, respectively, and //c// is a [|real number]. Using the identity |//u//|2 = //uu//* we can rewrite this as which is clearly a real quantity. This is because As another example, the expression where //a//, //b//, //c//, //x//, and //y// are real numbers, with //a// > 0 and //b// > 0, may be expressed in terms of the square of the [|absolute value] of a complex number. Define Then so

Geometric perspective
Consider completing the square for the equation Since //x//2 represents the area of a square with side of length //x//, and //bx// represents the area of a rectangle with sides //b// and //x//, the process of completing the square can be viewed as visual manipulation of rectangles. Simple attempts to combine the //x//2 and the //bx// rectangles into a larger square result in a missing corner. The term (//b///2)2 added to each side of the above equation is precisely the area of the missing corner, whence derives the terminology "completing the square". [|[1]]

A variation on the technique
As conventionally taught, completing the square consists of adding the third term, //v// 2 to to get a square. There are also cases in which one can add the middle term, either 2//uv// or −2//uv//, to to get a square.

Example: the sum of a positive number and its reciprocal
By writing we show that the sum of a positive number //x// and its reciprocal is always greater than or equal to 2. The square of a real expression is always greater than or equal to zero, which gives the stated bound; and here we achieve 2 just when //x// is 1, causing the square to vanish.

Example: factoring a simple quartic polynomial
Consider the problem of factoring the polynomial This is so the middle term is 2(//x//2)(18) = 36//x//2. Thus we get
 * __References__**
 * 1) Narasimhan, R. (2008). [|//Precalculus: Building Concepts and Connections//]. Cengage Learning. p. 133–134. [|ISBN] [|0-618-41301-4] . [] ., [|Section //Formula for the Vertex of a Quadratic Function//, page 133–134, figure 2.4.8]

= = =3__.Factorization__= Integers Main article: [|Integer factorization] By the [|fundamental theorem of arithmetic], every positive [|integer] greater than 1 has a unique [|prime factorization]. Given an algorithm for integer factorization, one can factor any integer down to its constituent [|primes] by repeated application of this algorithm. For very large numbers, no efficient [|algorithm] is known.

Polynomials
Main article: [|Factorization of polynomials]

Quadratic polynomials
Any [|quadratic polynomial] over the [|complex numbers] (polynomials of the form //a////x//2 + //b////x// + //c// where //a//, //b// , and //c// ∈ ) can be factored into an [|expression] with the form using the [|quadratic formula]. The method is as follows: where α and β are the two [|roots] of the polynomial, found with the [|quadratic formula].

Polynomials factorable over the integers
where and You can then set each binomial equal to zero, and solve for //x// to reveal the two roots. Factoring does not involve any other formulas, and is mostly just something you see when you come upon a quadratic equation. Take for example 2//x//2 − 5//x// + 2 = 0. Because //a// = 2 and //mn// = //a//, //mn// = 2, which means that of //m// and //n//, one is 1 and the other is 2. Now we have (2//x// + //p//)(//x// + //q//) = 0. Because //c// = 2 and //pq// = c, //pq// = 2, which means that of //p// and //q//, one is 1 and the other is 2 or one is −1 and the other is −2. A guess and check of substituting the 1 and 2, and −1 and −2, into //p// and //q// (while applying //pn// + //mq// = //b//) tells us that 2//x//2 − 5//x// + 2 = 0 factors into (2//x// − 1)(//x// − 2) = 0, giving us the roots //x// = {0.5, 2} If a polynomial with integer coefficients has a [|discriminant] that is a perfect square, that polynomial is factorable over the integers. For example, look at the polynomial 2//x//2 + 2//x// − 12. If you substitute the values of the expression into the quadratic formula, the discriminant //b//2 − 4//ac// becomes 22 − 4 × 2 × −12, which equals 100. 100 is a perfect square, so the polynomial 2//x//2 + 2//x// − 12 is factorable over the integers; its factors are 2, (//x// − 2), and (//x// + 3). Now look at the polynomial //x//2 + 93//x// − 2. Its discriminant, 932 − 4 × 1 × (−2), is equal to 8657, which is not a perfect square. So //x//2 + 93//x// − 2 cannot be factored over the integers.
 * Note:** A quick way to check whether the second term in the binomial should be positive or negative (in the example, 1 and 2 and −1 and −2) is to check the second operation in the trinomial (+ or −). If it is +, then check the first operation: if it is +, the terms will be positive, while if it is −, the terms will be negative. If the second operation is −, there will be one positive and one negative term; guess and check is the only way to determine which one is positive and which is negative.

Perfect square trinomials
A visual illustration of the identity (//a// + //b//)2 = //a//2 + 2//ab// + //b//2 Some quadratics can be factored into two identical binomials. These quadratics are called perfect square trinomials. Perfect square trinomials can be factored as follows: and

[[|edit]] Sum/difference of two squares
Main article: [|Difference of two squares] Another common type of algebraic factoring is called the [|difference of two squares]. It is the application of the formula to any two terms, whether or not they are perfect squares. If the two terms are subtracted, simply apply the formula. If they are added, the two binomials obtained from the factoring will each have an imaginary term. This formula can be represented as For example, 4//x//2 + 49 can be factored into (2//x// + 7//i//)(2//x// − 7//i//).

[[|edit]] Factoring by grouping
Another way to factor some polynomials is factoring by grouping. For those who like algorithms, "factoring by grouping" may be the best way to approach factoring a trinomial, as it takes the guess work out of the process. Factoring by grouping is done by placing the terms in the polynomial into two or more groups, where each group can be factored by a known method. The results of these factorizations can sometimes be combined to make an even more simplified expression. For example, to factor the polynomial Group similar terms, Factor out [|Greatest Common Factor], Factor out binomial

[[|edit]] AC Method
If a quadratic polynomial has rational solutions, we can find p and q so that pq = ac and p + q = b. (If the discriminant is a square number these exist, otherwise we have irrational or complex solutions, and the assumption of rational solutions is not valid.) The terms on top will have common factors that can be factored out and used to cancel the denominator, if it is not 1. As an example consider the quadratic polynomial: Inspection of the factors of ac = 36 leads to 4 + 9 = 13 = b.

Sum/difference of two cubes
Another formula for factoring is the sum or difference of two cubes. The sum can be represented by and the difference by For example, //x//3 − 103 (or //x//3 − 1000) can be factored into (//x// − 10)(//x//2 + 10//x// + 100).

Difference of two fourth powers
Another formula is the difference of two fourth powers, which is

Sum/difference of two fifth powers
Another formula for factoring is the sum or difference of two fifth powers. The sum can be represented by and the difference by

Sum/difference of two sixth powers
Then there's the formula for factoring the sum or difference of two sixth powers. The sum can be represented by and the difference by

Sum/difference of two seventh powers
And last there's the formula for factoring the sum or difference of two seventh powers. The sum can be represented by and the difference by

Difference of nth powers
This factorization can be extended to any positive integer power //n// by use of the [|geometric series]. By noting that and multiplying by the (//x// -1) factor, the desired result is found. To give the general form as above, we can replace //x// by //a/b// and multiply both sides by //b//n. This gives the general form for the difference of two //n//th powers as The corresponding sum of two //n//th powers depends on whether //n//is even or odd.If //n// is odd, //b// can be replaced by //-b// in the above formula. If //n// is even, the form is somewhat more tedious. for more information based on factorization go to [].