Factorization in Grade IX

Hello friends, today we are going to learn about the Factoring Expressions.

Factoring is just the reverse of expanding but a bit complex than a (b + c) = ab + ac, but the procedure followed is the same.

Factoring is generally an expression which is written as the product of two or more expressions.

For Example - 15 = 5 x 3, where 5 and 3 are both the factors of 15. The most basic ways to factor an expression is by finding the common factors.

Now what are common factors?

If each term of the expression has several factors and has atleast one factor that is common in them, then the common term is known is its common factor.

For Example – x² y⁴ + 2 x².

In this expression we can see that x² is common in both the terms. So it can be written as

x² ( y⁴ + 2 ). So now x² is the common factor of the above expression or it can also be said as it is the Greatest Common Factor( GCF) of the above expression.

The general way of writing the common factors is GCF x ( remaining expression).

We take another example for more understanding.

Let the expression be 8Y³B² + 16Y²B.

Now to find the common factors of this we have to see that what is common, so it seems like the term 8Y²B is common in the above expression. So it can be written as 8Y²B ( YB + 2 ). This is the same way as we have written the general formula GCF x ( remaining expression).

Now as I said above factoring is just opposite of expanding. So let's now verify this.

We take two terms ( x + 5) ( x + 1).

The expansion of this is x² + 6x + 5. And the factoring of this answer gives us ( x + 5) ( x + 1). So these are the factors of the given expression. We do this by splitting the x term in two parts. There are different ways of expanding but one among them helps us to find the common factor.

The 6x can be broken as 5x+1x, 2x+4x and 3x +3x. It satisfies x + 5x, so that we get our common factors. For doing this we have to follow some general rules.

Let the general quadratic equation be ax² + bx + c. So the general rules are.

1.multiply the constants and c without forgetting their signs.

2.Write down b, with its sign.

3.Write all the possible factors of the product of ac and find which pairs add up to give b .

4.now rewrite the x terms as the sum of two terms with these selected numbers as the coefficients

So by this we can find the factors of the quadratic expression. The above method we followed is known as the decomposition method.

To make perfect squares. Any expression of the form x² + 2ax + a² can be written as a perfect square (x + a)². To see whether an expression is a perfect square or not we first see that whether the constant term in it is a whole square or not, if it is a perfect square then take square root of it and multiply it by 2. If the result is equal to the coefficient of x then the term could be easily written as a perfect square.

For Example – Factorize x² + 8x + 16.

We can see that the constant term is 16, and it is a perfect square of 4.

Now 2 x 4 is equal to 8 which is the coefficient of x. So our given expression can be written as

(x + 4)².

Now we move to differences of square.

The differences of the square term are written in a general way as (something)² – (something)².

Let's understand it with the help of a simple example. Let's take (a²x² – b²), it can be factorised as (ax + b)(ax – b). We can see that the factors are identical except the sign between them.

For Example-

Factorize 9 – r².

To factorize it we use the above expression as the general function and find the value of a, x and b.

Here a² = 1, thus a = 1

x² = 9, thus x = 3

b² = r², thus b = r,

so the above expression can be written as ( 3 + r ) ( 3 – r ). To understand it more we will take another example.

Factorize 9p² – 4.

Here also we follow the above method and find the values of a, x and b.

a² = 9, thus a = 3x² = p², thus x = pb² = 4, thus b = 2,

so the above expression after factorization gives ( 3p + 2 ) ( 3p – 2 ).

There is one more thing that every expression cannot be factorized by this method, so we use the Quadratic Formula to find the roots and then we write the factors.

Now we move to the Difference of cubes.

It is very much similar to the difference of squares, the only difference is that it has cubes in the exponents in place of squares.

We write it in a general way as a³ – b³. Which can be factorized as (a - b)(a² - ab + b²). The factorization seems to be very difficult to remember, so what we have to remember is that if we see an expression like a³ – b³, then (a – b) is a common factor and the rest we can find out using the long division method.

Sum of Cubes.

The general way of writing this is a³ + b³. And when factorized we see that the result is

(a + b)(a² - ab + b²). In this also we have to remember only that when we see an expression like

a³ + b³, then we should remember that (a + b) is its common factor and the rest we can find out using the long division method.

Now as we have performed factorization on expressions with square and cube exponents, so now we move to the expressions with higher degrees. Sometimes students get panicked seeing higher degrees, but there is no need to panic as the expressions with higher degrees are as simple as the quadratic and cubic ones. There will always be a common factor in them because of which the remaining term will be converted into a quadratic expression.

For Example take (9x⁴ - 2x³ + 10x²) as your expression.

Now we can see that x² is common from the above expression. So we can finally write it as

x²(9x² - 2x + 10). Now the expression within the brackets can be solved by using the decomposition method. If we see that there is a constant term in it, then the simplest way is by using the long division method.

Let's understand this with the help of an example.

Solve x³ - 2x² - 5x + 6 with (x + 2) as one of its factors.

We will use the long division method to solve the above expression.

x² - 4x + 3

-------------------

(x + 2)| x³ - 2x² - 5x + 6)

x³ + 2x²

---------

-4x² - 5x

-4x² - 8x

---------

3x + 6

3x + 6

------

0

so the factor other than x+2 is x² - 4x + 3.

now we have to solve x² - 4x + 3 by using the decomposition method.

So now split -4x as -x and -3x. So we get the factors as (x - 1) and (x – 3).

so at the end the answer to x³ - 2x² - 5x + 6 is (x + 2)(x - 1)(x – 3).

In this question we were given one factor of the expression, what if there is no factor given in such type of questions. Then we could still factorize by assuming that ( x – k) as its common factor. Where k is some value which satisfies the given equation. If we find the value of k by hit and trial then we wil get our one factor and the rest could be done by long division method.

So let's understand this with the help of an example.

Solve x³ + 1.

So we have no common expression given with this question.

We assume the one factor as (x – k), now by hit and trial we can see that -1 satisfies the above equation so our factor is (x + 1). Now by using long division method we can find the rest.

x² - x + 1

-----------------

(x + 1)| x³ + 0x² + 0x + 1)

x³ + x²

-------

-x² + 0x

-x² - x

-------

x + 1

x + 1

-----

0

So the another factor is x² - x + 1.

Now the solution to x³ + 1 is equal to ( x + 1) (x² - x + 1).

So now I hope that you would be able to perform the operations on Factoring expressions

IX Grade Fractions

IX grade mathematics
Hello Friends today we all are going to understand the basic concept behind one of the most important and a bit complex of IX standard that is Fraction. Before proceeding further, let's talk about the basic concept behind Fractions. In simple mathematics a fraction is an expression that consists of numerator and denominator. For example a/b where a is the numerator and b is the denominator and both are integers. The most important condition is that denominator can never be a zero. Here we will try to understand some of the basic facts about fractions and also try to know the reason behind these facts.

Few of the facts about fractions are:

1. The integer “x” on the top of the fraction bar is the numerator and the integer “y” at the bottom is the denominator of the fraction.
   Fraction = numerator / denominator
   
2. Any of the integer can be considered as a fraction having a denominator equal to 1.
   X = X/1
   
3. If we are going to multiply numerator and denominator of a fraction with the same non zero factor does not manipulate the fraction. Basically in elementary school we classify it with the term equivalent fractions, but a better view is that equivalent fractions denote the same real number and need not be distinguished other than for simplicity.
   X/Y = CX/CY where C is not equal to zero.
   
4. A fraction of having the same denominator are added or subtracted by adding or subtracting the numerators.
   X/y + C/y = X + C / y
   X/y - C/y = X - C / y
   
5. If the fractions come with distinct denominators than firstly we need to turn them into fractions with the same denominators. This can be done by multiplying numerators and denominators with suitable factors, and then added using the rule above.
   X/y + c/d = Xd / yd + cy / yd = Xd + cy/ yd
   Similarly we can perform subtraction :
   X/y - c/d = Xd / yd - cy / yd = Xd - cy/ yd
   
6. In the case of multiplication, fractions are multiplied by multiplying numerators and multiplying denominators.
   X/Y * C/D = XC/YD
   
7. We can find the reciprocal of a fraction by switching numerators and denominators.
   
8. In division case : If we are going to divide a fraction by another fraction which is equivalent to multiplying with the reciprocal of the second fraction.
   X/Y divide C/D = XD / YC
   

Now lets go in detail with fractions. A fraction such as 5/4 is simplified in a kind of manner to explain the result of dividing 5 by 4. Commonly, A fraction X/Y explains the result of dividing X by Y. This explanation is true even if X and Y are not integers, an explanation of the principle that we make definitions in simple contexts and then generalize them so that all relevant rules remain true. However, if X and Y are not integer then the expression X/Y is called a ratio or quotient.
Division in a fraction can be defined as the solution of a multiplication problem.
Thus the fraction x = 5/4

is the solution of the equation
4x = 5
In more general form, the fraction
x = a/b

is the solution of the equation
bx = a.
The most important thing is to understand the basic definition as all of the above rules can be derived from it
Now let's see how to add fractions with different denominators:
Whenever we are going to add two fractions like 1/3 and 3/7 then we firstly need to convert them to fractions with the same denominator by applying the above rule number 3. Therefore we need to extract a common denominator. In a practical manner the more smaller the denominator, the more easier it is to manipulate the numbers and that time we mostly like to use the least common denominator. Meanwhile, the the product of the two denominators always works, and it is often, like in this case, also the least common denominator. So let's take this example to illustrate it in a more conventional manner:

1/3 = 1*7/3*7 = 7/21 and 3/7 = 3*3/7*3 = 9/21
Therefore:
1/3 + 3/7 = 7/21 + 9/21 = 7+9/21 = 16/21.


Now the next thing is how to multiply fraction :
Suppose we have two fractions that are x = 3/5 and y = 14/9
This shows that x and y satisfy the equations:
5x = 3 and 9y = 14
Now moving further and multiplying on both the sides of the first equation with 14 but we need to call 9y on the left hand side, we obtain
5x X 9y = 3 X 14
This can be further simplified as
45 xy = 42
which is the another way of explaining this:
xy = 42/45
Additionally we can further simplify this fraction by dividing numerator and denominator by 3 which gives us : 14/15


Now the next topic to illustrate is division of fractions:
Let's take the same above example and try to perform division operations with same fractions then:
a = x divide y = 3/5 divide 14/9
In this, division is the inverse procedure of multiplication, so it satisfies the equation
That shows as ay = x
or we can explain as
a X 14/9 = 3/5 We can find out the exact answer by multiplying on both sides of this equation with the reciprocal of y that is with 9/14
a X 14/9 X 9/14 = 3/5 X 9/14
This becomes : a * (14*9 / 9* 14) = 3*9/5*14
x divide y = a = 3*9 / 5*14 = 27/70.

 

Now the next topic to study is Greatest Common Factors and Least Common Multiples:
In simple mathematical manner a multiple of a natural number is the product of that number with another natural number. The most common example is, 6, 18, and 333 are all multiples of 3. Another thing to understand is that a factor of a natural number n is another natural number that divides n without remainder. Example to understand it, the factors of 12 are 1, 2, 3, 4, 6, 12. Similar concept is that a common factor of two natural numbers m and n is basically a number that is a factor of m and also of n. Let's take an example, 8 is a common factor of 32 and 48. Other common factors are 1, 2, 4, 8, 16.
Now we are going to understand the basic behind greatest common factor. The greatest common factor is just what the name suggests, so the greatest common factor of the above example that is 32 and 48 is 16. Similarly, a common multiple of two numbers is a multiple of both, and the least common multiple is the smallest common multiple. For example, 24, 120, 180 are all common multiples of 4 and 6, and 12 is the least common multiple.
The importance of this statement is that a common denominator of the two fractions is basically a common multiple of the two denominators and the least common denominator is generally the least common multiple of the two denominators.
The least common multiple (LCM) and the greatest common factor (GCF) of two numbers m and n are explained by the form:
LCM = m * n/ GCF
Let's take an example to understand is better, the Greatest common factor of 4 and 6 is 2 and their least common multiple is 12. So by using above formula we can formulate it:
12 = 4*6/2


Now the another area of study is Mixed Numbers or Mixed Fractions:
Mixed Numbers are basically fractions which are written as a natural number plus a fraction in which the denominator is always greater than the numerator. This area of study is important because the integer part in the fraction gives an indication of their sizes.
Mixed numbers are popular because the integer part gives an explanation of their size, but otherwise they have little to glorify them. They always form an exception (that is the only exception) to the rule that a unidentified or missing operator means multiplication, and they make the given arithmetic operations harder to carry out.


Now the another part of fractions are complex fractions. A complex fraction is a rational expression that has a fraction in its numerator, denominator or both. Example to show the complex fraction: a/b / c/d. The sign / denotes the division of two numbers or variables. Simplifying complex fractions involves following steps: firstly student needs to rewrite the numerator and denominator so that each form a single fraction. Now divide the numerator by the denominator by multiplying the numerator by the reciprocal of the denominator. In last step we need to simplify the rational expression.
So this is enough for today. In next class we all are going to learn the divisibility rules of the fractions and the use of Euclidean Algorithm while solving fractions.