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
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