Warm Up. Factor each quadratic. 1. x x + 24 = 0
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1 Warm Up Factor each quadratic. 1. x x + 24 = x 2 + 2x - 16 = x 2-6x = 0 4. x 2-49 = 0 Definition: 5.1 Polynomial Functions A polynomial function in one variable is a monomial or sum of monomials. All the exponents are positive integers. Coefficients are real numbers. EXAMPLES: 5x 2 3x 4 + 5x x 3 9x x 5 6x 4 2x 2 + x 9 Usually written in standard form...terms are in descending order of the exponents.
2 5.1 Polynomial Functions More Definitions: leading term...the first term in standard form; the term with the largest exponent leading coefficient...coefficient on leading term in standard form degree...the exponent on the leading term; the largest exponent EXAMPLES: Write each polynomial in standard form. Then state the leading term, the leading coefficient, the degree, and the constant. 1. 3x + 9x x 3 x + 5x x 5 + 9x Classify Polynomials by Degree: Degree Name Example PREVIOUS EXAMPLES: Classify each polynomial by degree. 1. 3x + 9x x 3 x + 5x x 5 + 9x
3 End Behavior of Polynomial Functions Look at these graphs. What do you observe about the ends if the degree is even? Odd? Positive Leading Coefficient EVEN DEGREE ODD DEGREE copyright purplemath.com End Behavior of Polynomial Functions Look at these graphs. What do you observe about the ends if the degree is even? Odd? Negative Leading Coefficient EVEN DEGREE ODD DEGREE copyright purplemath.com
4 Practice: End Behavior of Polynomial Functions REMEMBER! Most important thing to consider? Example 2. Describe the end behavior of 4x 4 + 2x 3 x + 7. Example 3. Describe the end behavior of 3x 2 + 6x x 3 2. Example 4. Describe the end behavior of 3x 7 + 5x copyright purplemath.com Summary: End Behavior of Polynomial Functions Most important thing to consider: the sign and degree of the leading term. When the leading coefficient is positive, the graph to the right. When the leading coefficient is negative, the graph to the right. When the function's degree is odd, the ends go in directions. When the function's degree is even, the ends go in directions.
5 Practice: End Behavior of Polynomial Functions REMEMBER! Most important thing to consider? Example 1. Which of the following could be the graph of a polynomial whose leading term is 3x 4? copyright purplemath.com Warm Up 1. Write in standard form. State the degree and name. f(x) = 3x + x 2 2x 4 + 5x 3 7 Name the function and describe the end behavior. 2. f(x) = x 5 + 2x 4 3x 3 + 2x 2 5x f(x) = 5 3x 2 + 4x 3 x 4 4. f(x) = 7 + 4x
6 Turning Points of Polynomial Functions points where the graph changes direction, the BUMPS. A linear function has no turning point. A quadratic function has only 1 turning point. A cubic function has at most 2 turning points. A quartic function has at most 3 turning points. Summary: A polynomial function has at most turning points.
7 PRACTICE: For each graph, determine the sign of the leading coefficient and the least possible degree of the polynomial function copyright purplemath.com PRACTICE: (no calculator) Write each polynomial in standard form. Name the function. State the degree and maximum number of turns. Describe the end behavior. Sketch the general shape. 1. f(x) = 5 x 2. f(x) = x 3 4x 3. f(x) = x 4 2x f(x) = 4x x f(x) = 3x 3 3x x f(x) = 6x + 1 x 3
8 Warm Up Name the function. State the maximum number of turns. Describe the end behavior. 1. f(x) = 2x 4 3x 3 + 2x 2 5x f(x) = 5 3x 2 + 4x 3 3. f(x) = 7 Summary: End Behavior of Polynomial Functions Most important thing to consider: the degree and sign of the leading term. Degree Positive Negative rises (up) on falls (down) right, on right, even rises (up) on left falls (down) on left odd rises (up) on right, falls (down) on left falls (down) on right, rises (up) on left
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Name: Class: Date: Algebra 2, Chapter 5 Review 4.4.1: I can factor a quadratic expression using various methods and support my decision. 1. (1 point) x 2 + 14x + 48 2. (1 point) x 2 x + 42 3. (1 point)
Notice that we are switching from the subtraction to adding the negative of the following term
MTH95 Day 6 Sections 5.3 & 7.1 Section 5.3 Polynomials and Polynomial Functions Definitions: Term Constant Factor Coefficient Polynomial Monomial Binomial Trinomial Degree of a term Degree of a Polynomial