Polynomial functions
- 2. A polynomial function is a function that can be written as: f(x)=axⁿ+bxⁿˉ¹+cxⁿˉ²…gx+h Where : a, b, c, …, g, and h are real numbers n is a positive integer.
- 3. POLYNOMIALS NOT POLYNOMIALS f(x)=½x²+2x-3 F(x)= ίxˉ² Why? Why not? Correct form ί is not a real number ½ , 2, and 3 are real -2 isn’t a positive number numbers 2 and 1 are both positive integers.
- 4. Standard form: f(x)=axⁿ+bxⁿˉ¹+cxⁿˉ²…gx+h Everything in the equation is multiplied together. Factored form: f(x)=(x-r1)(x-r2)(x-r3)…
- 5. Degree: the highest power of “x” in standard form. Ex. : f(x)=½x²+2x-3 Degree = 2 Leading coefficient: coefficient of the first term if the terms are in descending order. f(x)=½x²+2x-3 Leading Coefficient = ½
- 6. The factors of a polynomial are the quantities of the function when in factored form Ex: f(x)=(x-5)(x+7)(x-2) There are three factors in this polynomial. They are x-5, x+7, and x-2.
- 7. The zeros of a function are the numbers which can be inserted into “x” so that the function equals zero. Example: f(x)=(x-4)(x+7) f(x)=(x-4)(x+7) f(4)=(4-4)(4+7) f(-7)=(-7-4)(-7+7) f(4)=(0)(11) f(-7)=(-11)(0) f(4)=0 f(-7)=0 The zeros of this function are 4 and -7. When graphed, the zeros are the x-intercepts.
- 8. The number of zeros in a function are the same as the intercepts. Therefore, an equation with three different factors, such as f(x)=(x-2)(x+3)(x-7), would have three x- intercepts at (2,0), (-3,0), and (7,0). When two of the factors in an function are the same (ex.: f(x)=(x-2)(x+3)(x-2)), then the graph will, in this case, have two x-intercepts instead of three, with the curved line sitting on the x- axis instead of crossing it and intersecting in three places.
- 9. The graph of a function cannot intersect the x axis any more time than the number of factors it ha. When the factors of a function with just two factors are the same (ex.: f(x)=(x+3)(x+3)), the function has a “multiplicity of two.”