Polynomials:
"a polynomial is a monomial or the sum of monomials"
Classification of Polynomials:
Polynomials can be classified two different ways - by the number of terms and by theirdegree.
1. Number of terms.
1. Number of terms.
- A monomial has just one term. For example, 4x2 .Remember that a term contains both the variable(s) and its coefficient (the number in front of it.) So the is just one term.
- A binomial has two terms. For example: 5x2 -4x
- A trinomial has three terms. For example: 3y2+5y-2
- Any polynomial with four or more terms is just called a polynomial. For example: 2y5+ 7y3- 5y2+9y-2
2. Degree. The degree of the polynomial is found by looking at the term with the highest exponent on its variable(s).
Examples:
Examples:
- 5x2-2x+1 The highest exponent is the 2 so this is a 2nd degree trinomial.
- 3x4+4x2The highest exponent is the 4 so this is a 4th degree binomial.
- 8x-1 While it appears there is no exponent, the x has an understood exponent of 1; therefore, this is a 1st degree binomial.
- 5 There is no variable at all. Therefore, this is a 0 degree monomial. It is 0 degree because x0=1. So technically, 5 could be written as 5x0.
- 3x2y5 Since both variables are part of the same term, we must add their exponents together to determine the degree. 2+5=7 so this is a 7th degree monomial.
Sum and Difference of Cubes:
The sum and difference of cubes. These are the formulas:
a3 – b3 = (a – b)(a2 + ab + b2)
= (x – 2)(x2 + 2x + 22) = (x – 2)(x2 + 2x + 4)
= (3x + 1)((3x)2 – (3x)(1) + 12) = (3x + 1)(9x2 – 3x + 1)
= (xy2 – 4)((xy2)2 + (xy2)(4) + 42) = (xy2 – 4)(x2y4 + 4xy2 + 16) Standard Form and Factored Form:Dividing Polynomials:1. Synthetic Division
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Zeros and Multiplicity:3. Remainder Theorem:
Is (x-a) a factor of p(x)? If p(a) is not equal to zero then it is not a factor. If p(a) is equal to zero then it is a factor. |
2. Long Division