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polynomial / chapter 2 / class - 10 /Math / New St. Mery English School


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What is constant?

Ans:- A symbol having a fixed numerical value is called constant.

Ex:- 9,   -6,   4/7,   √2 etc 

What are variables?

Ans:- A symbol which may be assigned different numerical values is known as a variable.

Ex:- 2ㄫr , here 2ㄫ is constant but r is variable

What are Algebraic expressions?

Ans:- A combination of constants and variables connected by some or all of operations like +, -, x, ➗ is known as an algebraic expression.

Ex:- 5x2 + 9x + 4

         4x3 + √2x2 + 5/9x - 8

What is polynomial?

Ans:- An algebraic expression whose exponent is non-negative integer is called polynomial.

Division algorithm for polynomials

From Euclid's division lema , we know that 
Dividend = ( Divisor  Quotient ) + Remainder   
 
Applying the same to polynomial,
If  f(x) and g(x) are any two polynomial where g(x) ≠ 0 than we can find two other polynomials q(x) and r(x) such that
 
f(x) = q(x) ✕ g(x) + r(x)
where f(x) = dividend
           g(x) = divisor 
           q(x) = quotient
           r(x) = remainder
 
degree of f(x) ≥ degree of g(x)
degree of g(x) > degree of r(x)

If r(x) = 0 than the polynomial g(x) is a factor of polynomial f(x)
We can also say that polynomial f(x) is a multiple of polynomial g(x)

Type - 1
Q1- Apply division algorithm to find quotient q(x) and remainder r(x) on dividing f(x) by g(x) where
f(x) = 17x3 + 10x4 – 62x2 + 30x – 3 and g(x) = 1 + 2x2  + 7x
 
Solution- f(x) = 17x3 + 10x4 – 62x2 + 30x – 3        g(x) = 1 + 2x2  + 7x
                f(x) = 10x4 + 17x3 – 62x2 + 30x – 3        g(x) = 2x2  + 7x + 1
 
                   2x2  + 7x + 1  )   10x4 + 17x3 – 62x2 + 30x – 3  (  5x2 – 9x - 2   
                                               10x4 + 35x3 + 5x2 
                                             -_____-_____-_____
                                                          -18x3 - 67x2 + 30x – 3  
                                                           -18x3 - 63x2 - 9x 
                                                          +_____+____+_______
                                                                       - 4x2  + 39x - 3 
                                                                        - 4x2  - 14x - 2
                                                                        +____+____+___
                                                                                      53x - 1      
 
Q2- Find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in each of the following and verify division algorithm.
 
1-   f(x) =  8x3 + 6x4 + 17x2 + 21x + 7   ,     g(x) = 1 + 3x2 + 4x