Polynomial Functions Constant Coefficient Degree of a Polynomial Leading Coefficient X-Intercept Linear Factor Root of a Polynomial Even Function Odd Function Conjecture about degrees and zeros End Behavior of a function with an even exponent and a>0 End Behavior of a function with an even exponent and a<0 End Behavior of a function with an odd exponent and a>0 End Behavior of a function with an odd exponent and a<0 Turning Points y = 2x5 + 3x4 – 4x3 + 2x2 – x + 3 Constant Coefficient Degree of a Polynomial Leading Coefficient Maximum Number of Zeros End Behavior y = -x3 + 2x2 – 6x – 8 Constant Coefficient Degree of a Polynomial Leading Coefficient Maximum Number of Zeros End Behavior y = 3x4 – 5x3 – 12x2 To find the solutions, first……….. Next………. We use the factors to find the ……….. The x-intercept is _______________ and the roots are the _______________. With the equation above, first factor out____ to get______ Next ___________ to get__________. The factors are____________. The roots are_____________. The x-intercepts are____________. The maximum number of roots are______. But there are only______ because______ repeats. y = 12x4 – 79x3 + 106x2 + 81x - 36 = (4x+3)(3x-1)(x-3)(x-4) What is the constant of the function? What are the coefficients of the function? What is the degree of the function? What is the leading coefficient of the function? What are the factors of the function? What are the roots of the functions? What are the x-intercepts of the function? Describe the end behavior of the function. What is the maximum number of turning points in this function? How many turning points does this function have and what are they? Polynomial Functions Constant A constant is a number by itself with no variables. Coefficient A coefficient is the number in front of (being multiplied by) a variable or variables. Degree of a Polynomial and what does it tell you Leading Coefficient and what does it tell you It is equal to the greatest exponent of its variable. (the highest exponent). If it’s even the right and left side do the same thing, odd the right side and left side do the opposite. It is the number in front of the variable with the highest exponent. If it is positive, the right side of the graph rises, negative the right side of the graph falls. Also used to find max zeros and turning points. X-Intercept It is the coordinate of a point where a graph intersects the x-axis. Factor Something that multiplies to give you a product (in this case our equation) Root of a Polynomial Also known as the zero of the polynomial, it is the xvalue of the x-intercept and is the x-value that makes the equation equal to zero. Even Function A function that is symmetrical over the yaxis Odd Function A function that is symmetrical around the origin Conjecture about degrees and zeros The degree is the maximum number of zeros a function can have End Behavior of a function with an even exponent and a>0 Right and left side rise End Behavior of a function with an even exponent and a<0 Right and left side fall End Behavior of a function with an odd exponent and a>0 Right side rises, left side falls End Behavior of a function with an odd exponent and a<0 Right side falls, left side rises Turning Points The maximum number of turning points is one less than the degree y = 2x5 + 3x4 – 4x3 + 2x2 – x + 3 Constant 3 Coefficient 2, 3, -4, -1 Degree of a Polynomial 5 Leading Coefficient 2 Maximum Number of Zeros 5 End Behavior Right side rises, left side falls y = -x3 + 2x2 – 6x – 8 Constant -8 Coefficient -1, 2, -6 Degree of a Polynomial 3 Leading Coefficient -1 Maximum Number of Zeros 3 End Behavior Right side falls, left side rises y = 3x4 – 5x3 – 12x2 To find the solutions, first……….. First factor out any greatest common factors Next………. Factor the remaining terms if possible We use the factors to find the ……….. x-intercepts and roots The x-intercept is _______________ and the roots are the _______________. point where the line crosses the x-axis; x-values of the x-intercept/values that makes equation equal to 0 With the equation above, first factor out____ to get______ x2; x2(3x2 – 5x – 12) Next ___________ to get__________. (3x2 – 5x – 12); (x – 3)(3x + 4) The factors are____________. x2, x- 3, and 3x + 4 The roots are_____________. x = 0, 3, and – 4/3 The x-intercepts are____________. (0,0); (3, 0); and (-4/3, 0) The maximum number of roots are______. But there are only______ because______ repeats. 5; 3; 0 y = 12x4 – 79x3 + 106x2 + 81x - 36 = (4x+3)(3x-1)(x-3)(x-4) What is the constant of the function? -36 What are the coefficients of the function? 12, -79, 106, 81 What is the degree of the function? 4 What is the leading coefficient of the function? 12 What are the factors of the function? 4x+3; 3x – 1; x – 3; x - 4 What are the roots of the functions? -3/4; 1/3; 3, 4 What are the x-intercepts of the function? (-3/4, 0); (1/3, 0);(3,0); (4, 0) Describe the end behavior of the function. right side up, left side up What is the maximum number of turning points in this function? 3 How many turning points does this function have and what are they? 3; (-.3,-48.6); (1.7, 120.3);(3.6, -41.1)
