algebra 3 section r.4 polynomials
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Objectives of this Section Recognize Monomials Recognize Polynomials Add, Subtract, and Multiply Polynomials Know Formulas for Special Products. Algebra 3 Section R.4 Polynomials. - PowerPoint PPT PresentationTRANSCRIPT
Algebra 3 Section R.4Polynomials
Objectives of this Section• Recognize Monomials
• Recognize Polynomials• Add, Subtract, and Multiply
Polynomials• Know Formulas for Special Products
A monomial in one variable is the product of a constant times a variable raised to a nonnegative integer power. Thus, a monomial is of the form:
ax k
where a is a constant, x is a variable, and k > 0 is an integer.
Monomial Coefficient Degree
329
4xx
32
-9
Examples of Monomials
A polynomial in one variable is an algebraic expression of the form
a x a x a x ann
nn
11
1 0
Determine the coefficients and degree
of 2 3 54 2x x x .
Example:
Coefficients: 2, 0, -3, 1, -5
Degree: 4
Polynomials are added and subtracted by combining like terms.
Example: Addition
2 8 1 3 5 23 2 3x x x x x
2 3 8 5 1 23 3 2x x x x x
5 3 13 2x x x
Example: Subtraction
2 8 1 3 5 23 2 3x x x x x
2 8 1 3 5 23 2 3x x x x x
2 3 8 5 1 23 3 2x x x x x
x x x3 2 13 3
Polynomial multiplication can be done by using the distributive property multiple times.
Example: Multiplication
3 2 4 32x x x
3 3 4 3 3 2 2 4 2 32 2x x x x x x x
3 12 9 2 8 63 2 2x x x x x
3 14 17 63 2x x x
Special Product FormulasDifference of Two Squares
x a x a x a 2 2
Squares of Binomials, or Perfect Squares
x a x ax a
x a x ax a
2 2 2
2 2 2
2
2
Special Product Formulas
Miscellaneous Trinomials
x a x b x a b x ab 2
ax b cx d acx ad bc x bd 2
Cubes of Binomials, or Perfect Cubes
x a x ax a x a
x a x ax a x a
3 3 2 2 3
3 3 2 2 3
3 3
3 3
Special Product Formulas
Difference of Two Cubes
x a x a x ax a3 3 2 2
Sum of Two Cubes
x a x a x ax a3 3 2 2
Polynomials in Two VariablesThe degree of a polynomial in two variables is the highest degree of all the monomials with nonzero coefficients. The degree of each monomial is the sum of the powers of the variables.
DegreePolynomial2 32x y
2 43 4x xy-
3 2 43 2a ab b- +
Page 43: 1-6 all, 9-57 odds, 67-77 odds
Homework: