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| Monomial | An expression that is a number, a variable, or the product of a number and one or more variables. Also called a term. |
| Constant | Monomials that contain no variables. |
| Coefficient | The numerical factor in a monomial. |
| Degree of a Monomial | The sum of the exponents of its variables. The degree of a non-zero constant is zero. The constant zero has no degree. |
| Like Terms | Two monomials (terms) that are the same or differ only in their coefficients. |
What is not allowed in a monomial?
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Not a monomial, because there is a variable in the denominator |
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Not a monomial, because addition and subtraction give you a polynomial |
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Not a monomial, because a variable cannot be under a root sign. |
Examples to help with definitions:
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Definitions:
| Polynomial | An algebraic expression with more than one term (monomial). Polynomials with 1 term are called monomials. Polynomials with 2 terms are called binomials. Polynomials with 3 terms are called trinomials. |
| Degree of Polynomial | The sum of the degrees of its Dindividual terms (monomials). |
| Handshaking | Method of multiplying two polynomials. Each term of the first polynomial must be multiplied by each term of the second polynomial. |
| Like Terms should be combined (or simplified) using the distributive property. (3x+2x)=(3+2)x=5x Unlike Terms cannot be combined or simplified. |
Example 1:
Find the degree
of this polynomial:  |
Given Problem |
| 1st term: 8; 2nd term: 8; 3rd term: 9;
4th term: 2 |
Remember, the degree of a term is the sum of the exponents on the variables. |
| It is a 9th Degree Polynomial | Because the highest degreed term was 9. |
Example 2:
Add these polynomials:
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Given Problem |
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Rewrite the problem, using the commutative property of addition. Put like terms by each other. |
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Using the distributive property, group the like terms. |
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Simplify the expressions in the parentheses. |
Example 3:
Subtract these polynomials:
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Given Problem. |
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Rewrite the problem, using the distributive property. The subtraction must be carried out for each term of the second polynomial. |
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Rewrite the problem, using the commutative property of addition. Put like terms by each other. |
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Using the distributive property, group the like terms. |
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Simplify the expressions in the parentheses. |
Example 4:
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Given Problem. |
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First do all of the handshaking. I usually take the first term of the first polynomial, and multiply it by each of the terms of the 2nd polynomial, then repeat this with the remaining terms of the 1st polynomial. |
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Gather like terms. |
