when are the sum.deference,product and quotient of two monomials also a monomial?
The sum, difference, product, and quotient of two monomials will only result in a monomial in certain cases, which are as follows:
1. Sum and Difference:
The sum or difference of two monomials will always be a monomial if the two monomials have the same variables raised to the same powers. This means that the variables and their exponents must be identical for both monomials.
For example:
- The monomials 3x^2y and 4x^2y are like terms, and their sum (3x^2y + 4x^2y) is the monomial 7x^2y.
- However, the monomials 2xy and 3x^2y are not like terms, and their sum (2xy + 3x^2y) is not a monomial.
2. Product:
The product of two monomials will always be a monomial. This is because when multiplying monomials, the exponents of variables are added.
For example:
- The monomials 2x^3 and 3x^2 have a product of (2x^3) * (3x^2) = 6x^5, which is a monomial.
3. Quotient:
The quotient of two monomials will also result in a monomial if the divisor divides each term of the dividend with no remainder. In other words, all the exponents of the variables in the dividend must be equal to or higher than the corresponding exponents in the divisor.
For example:
- The monomials 20x^3 and 5x have a quotient of (20x^3) / (5x) = 4x^2, which is a monomial.
- However, the monomials 2xy and x are not divisible without a remainder, so their quotient (2xy) / (x) is not a monomial.
Remember, for the sum, difference, product, and quotient of two monomials to be a monomial, the variables and their exponents must either be the same or have the appropriate relationship described above.
To determine when the sum, difference, product, and quotient of two monomials will also be a monomial, we need to understand what a monomial is and examine the conditions for each operation.
A monomial is an algebraic expression that consists of a single term. It is typically written in the form of a coefficient multiplied by variables raised to non-negative integer exponents.
1. Sum of two monomials:
When adding two monomials, the resulting expression is still a monomial if and only if the variables and their corresponding exponents are the same in both monomials. For example, adding 3x^2 and 2x^2 will result in 5x^2, which is a monomial because the variables (x) and their exponents (2) are the same.
2. Difference of two monomials:
Similar to the sum, subtracting two monomials will result in a monomial if and only if the variables and their corresponding exponents are the same in both monomials. For example, subtracting 5x^3 from 7x^3 will give us 2x^3, which is a monomial because the variables (x) and their exponents (3) are the same.
3. Product of two monomials:
When multiplying two monomials together, the result will always be a monomial. The variables are combined by adding their exponents, and the coefficients are multiplied. For instance, multiplying 2x^2 by 3x^3 will yield 6x^5, which is a monomial.
4. Quotient of two monomials:
For the quotient of two monomials to be a monomial, the divisor must not contain any variables that are not present in the numerator, and the exponents of the variables in the divisor must be equal to or smaller than the exponents of the corresponding variables in the numerator. For example, dividing 10x^4 by 2x^2 will result in 5x^2, which is a monomial because the variables (x) and their exponents (2) meet the criteria.
Therefore, the sum, difference, product, and quotient of two monomials are also monomials in specific situations, as outlined above.
a monomial has just one term, such as x^2, 3, or 3y
So, if A and B are monomials,
the product is always another monomial
the sum or difference A+B or A-B is also a monomial as long as A and B are both in the same variable and same power. That is, 2x+2x = 4x and 6y-2y = 4y, but 2x+3z is a binomial, since the x and z terms cannot be combined. And 4x^2+5x is also a binomial, since the powers are not the same.
Division is also more complicated. A/B is a monomial if A and B both involve the same variable, and the power of A is not less than the power of B. 6x^3/2x = 3x^2 which is a monomial (as long as x ≠ 0, because division by zero is undefined).
6x/2x^2 = 3/x which is also not a monomial, since polynomials and monomials involve non-negative powers of the variable.
read the article in wikipedia on moomials