Explanation:
A monomial is a single number or a number multiplied by any number of variables.
Examples of monomials:
1 (a single number)
x (a number, 1, multiplied by one variable, x; although the 1 is not written the re is always a 1 preceding a variable unless there is an actual number written there.)
3x^2 (a number, 3, multiplied by x and multiplied again by x since x * x = x^2)
-5xy (a number, 5, multiplied by two variables, x and y)
A polynomial is a sum of monomials. When you add monomials together, you get a polynomial.
Examples of polynomials:
4x - 3
5x^2 + 2xy + 2x
5z^4 + 4z^3 - 2z^2 + 5z - 32
Other terms associated with monomials and polynomials:
The degree of a monomial is the sum of the exponents of all the variables of the monomial. Keep in mind that x is the same as x^1, so its exponent is 1.
Examples of degree of monomial:
4x^2 degree is 2
-5x^2y degree is 3 (2 from the x plus 1 from the y)
-23x^5yz degree is 7 (5 from the x plus 1 from y plus 1 from z)
The degree of a polynomial is the degree of its monomial with highest degree. In other words, find the degree of each monomial of the polynomial. Choose the highest degree as the degree of the polynomial.
Examples of degree of polynomial:
x^2 + 2x - 6 degree is 2 (the x^2 term has the highest degree, and it is 2)
4x^3y^3 + 2x^4y - 4x degree is 6 (the degree of 4x^3y^3 is 6; the degree of 2x^4y is 5; the degree of -4x is 1; the highest degree of any term is 6, so the degree of the polynomial is 6)
8x^6 + 9y^6 degree 6 (both monomials have degree 6, so the highest degree is 6, and that is the degree of the polynomial)
Each monomial that is part of a polynomial is called a term.
Example of number of terms:
This polynomial has 3 terms: x^2 + 2x - 5
In a polynomial, all terms are added together. The only operation that exists between terms forming a polynomial is addition. If you see a subtraction, it is just because it's a short way of writing plus a negative term.
Examples of addition of terms:
5x^2 + 2 This polynomial has two terms: 5x^2 and 2
8x - 2 This is the same as 8x + (-2) This polynomial has two terms: 8x and -2. Notice that the second term is -2. Just because the polynomial was written as 8x - 2, do not think that the terms are 8x and 2, and there is a subtraction between them. The correct way of thinking is that the terms are 8x and -2, and they are being added since terms are always added.
Coefficients and variables: the number part of a term is called a coefficient, The variable part of a term is called the variable part. The variable part includes all variables and exponents of the term.
Examples of coefficients and variable parts:
3x^2 3 is the coefficient; x^2 is the variable part
-6xy -6 is the coefficient; xy is the variable part
Name the coefficients of this polynomial:
5x^3 - 6x^2 + 4x - 8
Answer: 5, -6, 4, -8
Constant term: The term of a polynomial that is only a number and has no variable is called the constant term.
Example of constant terms:
6x^2 + 3x - 8 -8 is the constant term
-2x^2 + 12 12 is the constant term
Ok, I'll stop here before this turns into a book. If you have questions, ask in the comments.
