1. Polynomial Functions Chapter 2 Part 1

2. Standard Form f(x)=ax 2 +bx+c Vertex Form f(x)=a(x-h) 2 +k Intercept Form f(x)=a(x-d)(x-e) y-int (0, c) let x = 0 then solve for f(x) let x = 0 then solve for f(x) x-int(s) let f(x) = 0 then factor or use quadratic formula let f(x) = 0 then solve for x (d, 0) & (e, 0) Vertex y v = substitute x v and solve for y (h, k) y v = substitute x v and solve for y Quadratic Functions Quadratic Formula Parabola opens up if a > 0 (pos); opens down if a < 0 (neg) Parabola is symmetrical about the vertical line passing through the vertex Parabola opens up if a > 0 (pos); opens down if a < 0 (neg) Parabola is symmetrical about the vertical line passing through the vertex

3. Features of Polynomial Graphs Polynomial graphs are continuous Degree of the polynomial (n) ◦ n = the largest exponent End Behavior y-intercept: ◦ let x = 0 then evaluate for y Zero(s) (aka x-intercepts) of the function: ◦ let y or f(x) = 0 then solve for x Turning Points (aka Relative Extrema) : ◦ estimate (algebraic method will be learned in Calculus)

4. Positive Lead CoefficientNegative Lead Coefficient Odd Exponent Think: y = x Down on left; Up on right f(x) ⇒ - ∞ as x ⇒ - ∞ f(x) ⇒ + ∞ as x ⇒ + ∞ Think: y = - x Up on left; Down on right f(x) ⇒ + ∞ as x ⇒ - ∞ f(x) ⇒ - ∞ as x ⇒ + ∞ Even Exponent Think: y = x 2 Up on both sides f(x) ⇒ + ∞ as x ⇒ - ∞ f(x) ⇒ + ∞ as x ⇒ + ∞ Think: y = - x 2 Down on both sides f(x) ⇒ - ∞ as x ⇒ - ∞ f(x) ⇒ - ∞ as x ⇒ + ∞ Leading Term Test (aka End Behavior)

5. Zeros and Turning Points Degree of the polynomial (n) ◦ n = the largest exponent Maximum # of zeros = n ◦ zeros with odd multiplicity pass through x-axis ◦ zeros with even multiplicity turn at x-axis Maximum # of turning points = n-1

6. Long Division of Polynomials 8657÷21(8x 3 +6x 2 +5x+7)÷(2x+1) 1.Divide 2.Multiply 3.Subtract 4.Repeat 5.+ Remainder Divisor 1.Divide first term inside by first term outside 2.Multiply answer by divisor 3.Subtract by changing to “add opposite” 4.Repeat until degree inside < degree outside 5.+ Remainder Divisor

7. Synthetic Division of Polynomials (a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 ) ÷ (x – k) coefficients of dividend k b 3 b 2 b 1 b 0 r coeff of quotient degree of quotient is one less than degree of dividend Interpreting the results: a 4 a 3 a 2 a 1 a 0 remainder Vertical: Add Diagonal: multiply by k

8. Remainder and Factor Theorems Remainder Theorem ◦ If f(x) is divided by (x-k) then f(k) = r Factor Theorem ◦ (x-k) is a factor of f(x) if and only if f(k) = 0

9. Zeros of Polynomial Functions If a is a zero of f(x) then: x=a is a solution of f(x) = 0 (x – a) is a factor of f(x) (a, 0) is an x-intercept on the graph of f(x) ◦ if a is a real number

10. Rational Zero Test If f(x) is a polynomial function f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 with integer coefficients, then all rational zeros of f(x) will have the form: p = ± factors of constant term (a 0 ) q ± factors of leading coefficient (a n )

11. Fundamental Theorem of Algebra A polynomial of degree n has exactly n complex zeros (including repeated zeros). Complex Conjugate Theorem Complex zeros of a polynomial function occur in conjugate pairs. In other words: if (a + bi) is a zero then (a – bi) is also a zero

12. Upper and Lower Bound Tests Upper Bound Test ◦ Works only for positive values of k ◦ Indicates where the graph has reached its “end behavior” on the right (no need to test larger k values) ◦ Signs of Quotient and Remainder are all pos or all neg Lower Bound Test ◦ Works only for negative values of k ◦ Indicates where the graph has reached its “end behavior” on the left (no need to test larger negative k values) ◦ Signs of Quotient and Remainder alternate between pos and neg Zero can be thought of as either pos or neg

13. Descartes’ Rule of Signs If f(x) is a polynomial function with real coefficients, then ◦ the number of positive real zeros is equal to the number of sign changes in f(x) or less than that by an even number. ◦ the number of negative real zeros is equal to the number of sign changes in f(-x) ◦ or less than that by an even number.