A polynomial is an algebraic expression with literal. Mostly we use x, y literal in polynomials. Literals are also called as variables. In polynomial, the variables have only positive integral exponents.
For example, 15 + 2x + x2 , 7x3 + 5xy2 + 12 y3 .
The first example is polynomials in one variable x.
The second example is polynomials in two variables x and y.
Polynomials having only one term are known as monomials. Monomial is one of the type of polynomials. For example,
x3 , x2 x , y5, 5x5, 6y3 .
Let us practice division of a monomial problems.
Divide monomial problems for Practice:
There are two rules for dividing a monomial by a monomial. They are following,
Rule 1: The coefficient of the quotient of two monomials is equal to the quotient of the coefficients of the monomials in question.
Rule 2: In the quotient of two monomials, the variable part is equal to the quotient of the variable parts in the monomials in question.
Problem 1:
Divide by x5 by x3 monomials.
Solution:
x5 ÷ x3 = `x^5/x^3 `
= x2 .
Problem 2:
Divide by 15x5 by 5x4 monomials.
Solution:
15x5 ÷ 5x4 = `(15x^5)/(5x^4) `
= 5x .
Problem 3:
Divide by -20x4 by 10x monomials.
Solution:
- 20x4 ÷ 10x = `(-20x^4)/(10x) `
= `(-20/10)`` (x^4/x)`
= - 2 x3 .
Problem 4:
Divide by 3y3 by `sqrt(3)` y monomials.
Solution:
3y3 ÷ `sqrt(3)` y = `(3y^3)/(sqrt(3)y) `
= `(3/sqrt(3))`` (y^3/y)`
= `sqrt(3)` y2.
Problem 5:
Divide by 2x2 by 2x monomials.
Solution:
2x2 ÷ 2x = `(2x^2)/(2x) `
= `(2/2)`` (x^2/x)`
= x .
Problem 6:
Divide by -3x3 by x2 monomials.
Solution:
- 3x3 ÷ x2 = `(-3x^3)/(x^2) `
= `(-3/1)`` (x^3/x^2)`
= - 3 x .
Practice Problems on monomials:
Practice Problem 1:
Divide by `2/3` x2 by x monomials.
Answer:
`2/3` x .
Practice Problem 2:
Divide by `sqrt(5)` x4 by 5x3 monomials.
Answer:
`x/sqrt(5)` .
Practice Problem 3:
Divide by `sqrt(3)` a3 by 2a monomials.
Answer:
`(sqrt(3)a^2)/2` .
Practice Problem 4:
Divide by 4a4 by - 2 `sqrt(2)` a2 monomials.
Answer:
`-sqrt(2)a^2` .
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