Chapter 5 Notes.pptx - 5.7 Factoring Equations 5.5 Special...

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5.5 Special Types of Factoring 5.4 Factoring Trinomials 5.3 Factoring Polynomials 5. 2 Multiplication of Polynomials 5.1 Polynomial Functions 5.7 Factoring Equations Chapter 5 – Polynomial Expressions and Functions
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5.5 Special Types of Factoring 5.4 Factoring Trinomials 5.3 Factoring Polynomials 5. 2 Multiplication of Polynomials 5.1 Polynomial Functions 5.7 Factoring Equations Chapter 5 – Polynomial Expressions and Functions
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5.1 Polynomial Functions A polynomial is a finite sum of terms of the form ax n , where a is a real number and n is a whole number. For example, A term is a number, a variable, or a product of numbers and variables raised to a power. In a polynomial, terms are separated by + or signs. The polynomial above is made up of 5 terms. Identifying Monomials A monomial is a polynomial with exactly one term. Monomials: Not Monomials: Variable in the denominator Exponent is negative x x x x 4 3 2 3 5 6 2 3 x 3 2 2 1 bc a 7 y 2 3 6 x
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5.1 Polynomial Functions Writing Monomials Write a monomial the represents the volume of three cubes with side lengths, x. x x x Volume The volume of a cube is x · x · x = x 3 . So the cubes will have a volume of 3x 3 which represents a monomial. The degree of a polynomial The monomial (or term) with the highest degree (or number of variables). 2 nd Degree Trinomial 1 st Degree Binomial 6 th Degree Polynomial 1 3 7 2 x x 6 5 x 5 3 4 2 3 2 4 3 y xy y x x
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5.1 Polynomial Functions Polynomials may contain different variables. However, if a polynomial has only one variable, it is called the polynomial of one variable . The leading coefficient of a polynomial is the coefficient of the monomial, or term with the greatest degree. The degree of the polynomial is the term with the largest number of variables. Polynomials can be named according to the degree. A polynomial degree 1 is a linear polynomial , a polynomial of degree 2 is a quadratic polynomial , and a polynomial of degree 3 is a cubic polynomial . Polynomial Degree Leading Coefficient Type -63 0 -98 Constant 2x – 7 1 2 Linear -5z + 9z 2 + 7 2 9 Quadratic -2x 3 + 4x 2 + x – 1 3 -2 Cubic 7 – x + 4x 2 + x 5 5 1 5 th degree The table below classifies one variable polynomials by degree and type. It also identifies the leading coefficient.
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Adding Polynomials Polynomials can be added using the commutative and associative properties and then combine like terms. Combine Like Terms Group Like Terms 5.1 Polynomial Functions Polynomials can also be added vertically by placing like terms in the same columns and adding each column. Align like terms according to the variable and degree and add only the coefficients . 3 10 6 3 ) 6 4 ( ) 5 ( ) 6 5 ( ) 3 4 ( 2 2 2 2 2 x x x x x x x x x x 5 5 4 to added 4 3 2 3 3 y y y y 9 3 5 5 5 0 5 4 4 3 0 2 3 2 3 2 3 y y y y y y y y y
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Opposites of Polynomials Polynomials Opposite 2x – 4 -2x + 4 -x 2 – 2x + 9 x 2 + 2x – 9 6x 3 – 12 -6x 3 + 12 -3x 4 – 2x 2 – 8x + 3 3x 4 + 2x 2 + 8x - 3
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