CLASS-6
DIVISION OF ALGEBRAIC EXPRESSION OF POLYNOMIAL BY MONOMIAL
DIVISION OF ALGEBRAIC EXPRESSION OF POLYNOMIAL BY MONOMIAL -
When dividing an algebraic expression of a polynomial by a monomial (a single-term polynomial), you can use the distributive property of division to simplify the expression. Here's how to do it step by step:
Let's say you want to divide the polynomial P(x) by the monomial M(x):
P(x) = ax³+ bx²+ cx + d
M(x) = mxⁿ (where n is a non-negative integer)
The division can be expressed as:
P(x)
----------
M(x)
Here's how to perform the division:-
- Divide each term of the polynomial P(x) by the monomial M(x):
ax³ bx² cx d
-------- + -------- + -------- + -------
mxⁿ mxⁿ mxⁿ mxⁿ
2. Simplify each term individually by dividing the coefficients and subtracting the exponents:
a b c d
------- x³ˉⁿ + ------- x²ˉⁿ + ------- x¹ˉⁿ + ------- xˉⁿ
m m m m
3. Combine like terms, if any, by adding or subtracting them based on the exponents:The terms with the same exponent can be combined, but the terms with negative exponents should be rewritten as fractions:
a b c d
------- x³ˉⁿ + ------- x²ˉⁿ + ------- x¹ˉⁿ + ------- xˉⁿ
m m m m
4. If there are any terms with negative exponents (xˉⁿ), rewrite them as fractions with positive exponents:
a b c d 1
------- x³ˉⁿ + ------ x²ˉⁿ + ------- x¹ˉⁿ + ------ . ------
m m m m xⁿ
Now, the expression is fully simplified after dividing the polynomial P(x) by the monomial M(x). Each term in the resulting expression has been divided by the monomial, and any negative exponents have been addressed as fractions.
Example.1) Divide the polynomial P(x) = 4x³− 6x²+ 10x − 12 by the monomial M(x) = 2x.
Ans.)
P(x)
We want to find ---------
M(x)
- Divide each term of the polynomial P(x) by the monomial M(x):-
4x³ 6x² 10x¹ 12 x⁰
-------- - -------- + --------- - ---------
2x 2x 2x¹ 2x
2. Simplify each term individually by dividing the coefficients and subtracting the exponents:
4 6 10 12
------- x³ˉ¹- ------- x²ˉ¹ + ------- x¹ˉ¹ - ------- x⁰ˉ¹
2 2 2 2
3. Combine like terms:-
2 3 5 6
------- x² - ------- x¹ + ------- x⁰ - ------- xˉ¹
1 1 1 1
4. Simplify further:-
2x² - 3x + 5 - 6 xˉ¹
5. Rewrite the term with a negative exponent (xˉ¹) as a fraction with a positive exponent:
6
2x² - 3x + 5 - ------
x
So, the result of dividing the polynomial 4x³− 6x²+ 10x − 12 by the monomial 2x is:
6
2x² - 3x + 5 - ------ (Ans.)
x
Example.2) Divide the polynomial P(x) = 18x⁴ − 10x²+ 6x⁷ by the monomial M(x) = 2x².
Ans.)
Example.3) Divide the polynomial P(x) = 5xy²− 12x³y⁴ - 6 by the monomial M(x) = 3x.
Ans.)
Example.4) Divide the polynomial P(x) = 12x⁵ − 9x³+ 3x² by the monomial M(x) = 3x².
Ans.)
