Hello friends today we are going to discuss about factoring of two degree and three degree polynomials which you need to study in grade XII of gujarat secondary education board. In earlier class you must have gone through about the basics of polynomials and For more practice you can use properties worksheets . We are here to study about 2nd degree and 3rd degree polynomial. Now let us discuss about the 2nd degree polynomial.
2nd degree polynomial:
The degree of polynomial represent the highest degree on non zero term which contain variable.
2nd degree polynomial:
It means its variable has highest power of 2.
Example -2X2+2y2+9=0
We can do fallowing operations on 2nd degree polynomial.
Addition
Subtraction
Multiplication To know more about Two Degree Polynomials click here,
Now let us discuss Addition of polynomial with the help of fallowing examples
Example: 2X2+2Y2+2Z2 + 3X2-Y2-Z2
For solving this type of problem we just need to use the property of associative and distributive law which you have studied in junior classes.
(2X2+2Y2+2Z2)+( 3X2-Y2-Z2)
2X2 +3X2 +2Y2-Y2 +2Z2-Z2
(2X2 +3X2)+ (2Y2-Y2)+(2Z2-Z2)
5X2+Y2+Z2
Same things we can apply for subtraction but for more clearance we see one example
Example: Subtract 2X2+4Y2+6Z2 from 4X2-6Y2-4Z2
(4X2-6Y2-4Z2)-( 2X2+4Y2+6Z2 )
4X2-2X2-6Y2-4Y2-4Z2-6Z2
2X2-10Y2-10Z2
After addition and subtraction we move to multiplication by this process we can change the degree of polynomial. How to multiply we see by example given blow
2X2+2Y2+2Z2 multiply by 3X2-Y2-Z2
(2X2+2Y2+2Z2)* (3X2-Y2-Z2)
(2X2*3X2)+( 2Y2 *-Y2)+( 2Z2 *-Z2)
6X4-2Y4-2Z4
As you have seen in the above example that degree of polynomial can be changed by the multiplication in the above problem firstly the degree of polynomial was 2 now it is changed to 4
Division can be also done by the polynomial that you will be going to study in further class. Now I think we have well understood about 2nd degree polynomial now let’s move to 3rd degree of polynomial as it is more important for you .Now you came to know that if polynomial is of 3rd degree then its highest power will be 3.Like 2nd degree we can implement same operation on.
Factoring 3rd degree polynomials:
All algebraic expressions one or more than variables that are called polynomial .it terms are not in negative exponent.
Polynomial in algebraic expression P(x) =a0+a1x+a2x2……..an-1 xn-1+anxn, where a0, a1,…an.and are real number and n is non negative integer .and in polynomial x real degree of n.
A polynomial of degree is in three term like ax3+bx2+cx+d, a? 0 that is also called cubic polynomial.
Some step for third degree polynomials
And this value is satisfying this polynomial...
: Let’s check my answer.
First you plug the value in given polynomial.
P(x) =x3+7x2+12x
Put x= -4
P (-4) = (-4)3+ 7(-4)2+ 12(-4)
P (-4) = -64+ 122 - 48
P (-4) =0
Here -4 values satisfy this polynomial. It means -4 is factor of this polemical.
In this way you can check your all value by this process.
In this section only these parts are important and You can also refer grade XI blog for further reading on Polynomials in Grade XI.Read more maths topics of different grades such as Angles of triangles and polygons in the next session here.
2nd degree polynomial:
The degree of polynomial represent the highest degree on non zero term which contain variable.
2nd degree polynomial:
It means its variable has highest power of 2.
Example -2X2+2y2+9=0
We can do fallowing operations on 2nd degree polynomial.
Addition
Subtraction
Multiplication To know more about Two Degree Polynomials click here,
Now let us discuss Addition of polynomial with the help of fallowing examples
Example: 2X2+2Y2+2Z2 + 3X2-Y2-Z2
For solving this type of problem we just need to use the property of associative and distributive law which you have studied in junior classes.
(2X2+2Y2+2Z2)+( 3X2-Y2-Z2)
2X2 +3X2 +2Y2-Y2 +2Z2-Z2
(2X2 +3X2)+ (2Y2-Y2)+(2Z2-Z2)
5X2+Y2+Z2
Same things we can apply for subtraction but for more clearance we see one example
Example: Subtract 2X2+4Y2+6Z2 from 4X2-6Y2-4Z2
(4X2-6Y2-4Z2)-( 2X2+4Y2+6Z2 )
4X2-2X2-6Y2-4Y2-4Z2-6Z2
2X2-10Y2-10Z2
After addition and subtraction we move to multiplication by this process we can change the degree of polynomial. How to multiply we see by example given blow
2X2+2Y2+2Z2 multiply by 3X2-Y2-Z2
(2X2+2Y2+2Z2)* (3X2-Y2-Z2)
(2X2*3X2)+( 2Y2 *-Y2)+( 2Z2 *-Z2)
6X4-2Y4-2Z4
As you have seen in the above example that degree of polynomial can be changed by the multiplication in the above problem firstly the degree of polynomial was 2 now it is changed to 4
Division can be also done by the polynomial that you will be going to study in further class. Now I think we have well understood about 2nd degree polynomial now let’s move to 3rd degree of polynomial as it is more important for you .Now you came to know that if polynomial is of 3rd degree then its highest power will be 3.Like 2nd degree we can implement same operation on.
Factoring 3rd degree polynomials:
All algebraic expressions one or more than variables that are called polynomial .it terms are not in negative exponent.
Polynomial in algebraic expression P(x) =a0+a1x+a2x2……..an-1 xn-1+anxn, where a0, a1,…an.and are real number and n is non negative integer .and in polynomial x real degree of n.
A polynomial of degree is in three term like ax3+bx2+cx+d, a? 0 that is also called cubic polynomial.
Some step for third degree polynomials
1. In numerical ,plug x=1,-1,-2,2 etc in P(x)
2. If p (1) =0 then x-1 is factor of p(x) polynomial .
3. If p(x) is a cubic ,then it divide it by x-1
4. Now factorize quadratic polynomial.
Now lets the example of three degree
Ex: x3+7x2 +12x
Sol: let P(x) =x3+7x2 +12x
Let’s solve in simple form.
P(x) =x(x2+7x+12)
P(x) =x(x2+4x+3x+12)
P(x) =x(x(x+4) +3(x+4))
P(x) =x(x+4) (x+3)
This value is factor of this polynomial mean x=0, x=-4, x=-3And this value is satisfying this polynomial...
: Let’s check my answer.
First you plug the value in given polynomial.
P(x) =x3+7x2+12x
Put x= -4
P (-4) = (-4)3+ 7(-4)2+ 12(-4)
P (-4) = -64+ 122 - 48
P (-4) =0
Here -4 values satisfy this polynomial. It means -4 is factor of this polemical.
In this way you can check your all value by this process.
In this section only these parts are important and You can also refer grade XI blog for further reading on Polynomials in Grade XI.Read more maths topics of different grades such as Angles of triangles and polygons in the next session here.
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