Given zero, find other zeros.

Given zero, find other zeros. Polynomial Information Parabola Rational Functions\Asymptotes Word Problem Complex Numbers Click on buttons to go to a topic. Not all topics are covered here, but this should help you study. Click home buttons on the bottom right of each page to come back to this screen. If there is an error or a question, please notify me by e-mail or AIM. Writing Equations given zeros Intermediate Value Theorem \ Bounds Inequalities Write Equation Given a Sketch Rational Root Theorem

Given zeros, find other zeros. 1) Use synthetic to find the depressed equation given a zero. 2) If first zero was complex, use the conjugate to find the depressed equation again. 3) Reduce until you reach a quadratic, then factor or use quadratic formula to find the other zeros.

Parabolas • Write the equation of the formula in vertex form using completing the square! • State the vertex and the axis of symmetry • Which way does it open and why? • State the intercepts • Describe the transformation and graph, state all key points • State the range • State the intervals of increase and decrease Intervals of increase and decrease, use x-values for interval notation. Use parenthesis. Remember, range is y-values, and include the vertex. Remember the ‘x =‘ for axis of symmetry! You only factor the x2 and x term Remember to balance the equation. x – intercept: y = 0 y – intercept: x = 0

Word Problem

Writing equations given zeros Clear Clear Box Work Distribute Work

Intermediate value theorem, bounds. Intermediate value theorem: Given a continuous function in the interval [a,b], if f(a) and f(b) are of different signs, then there is at least one zero between a and b. Regarding bounds, just understand the application of the formula and what it means. Remember, in using the formula, you don’t use the leading coefficient. Bounds. Let f denote a polynomial function whose leading coefficient is 1. A bound M on the zeros of f is the smaller of the two numbers: Max{1, |a0| + |a1| + .. + |an-1|}, 1 + Max{|a0|, |a1|, .. |an-1|} f(3) = -9 f(4) = -7 f(5) = -3 f(6) = 3 There is a zero in the interval [5,6] because there is a sign change, and by intermediate value theorem, a zero must exist in that interval.

Inequalities • Move Polynomial so that f is on the left side, and zero is on the right side. Write as a single quotient (Common denominator) • Determine the numbers where f equals zero or is undefined. • Use those values to separate the real number line. (open and closed) • Select a number in each interval and evaluate. • If f(x) > 0, all x’s in interval are greater than zero. • If f(x) < 0, all x’s in interval are less than zero. 1 -1 -3 -2 -4 2 0 Closed, equals to and it’s a zero. We want greater than or equal to zero, so use the POSITIVE intervals and use open and closed circles to decide [ ] or ( ) Open, even though it’s equals to, it’s undefined there, so x can’t equal 1 or -1. + – + – [ -3, -1) U (1, ∞)

Write Equation given a sketch Remember: Cross is odd multiplicity. Touch is even multiplicity. -4 -2 1 4

If all coefficients are integers, list all possible combinations. Remember, p are all the factors of the constant, q are all the factors of the leading coefficient. • Test each possible zero until you find a factor. A factor HAS A REMAINDER OF ZERO! • Repeat the process until you get a quadratic or a factorable equation. • Find other zeros by quadratic formula or factoring. • Note: If all coefficients are not integers, you cannot use rational root theorem. In this case, it will most likely be a calculator problem where you will estimate and use those zeros. • Note 2: It’s possible for a zero to repeat, remember that. Rational Root Theorem 1 1 -3 -1 13 -10 1 -2 -3 10 1 -2 -3 10 0 -1 1 -2 -3 10 -1 3 0 1 -3 0 10 2 1 -2 -3 10 2 0 -6 1 0 -3 4 -2 1 -2 -3 10 -2 8 -10 1 -4 5 0

Polynomial Information Degree, end behavior, parent function. X and Y intercepts Cross or Touch at x-intercepts. Determine max\min with calculator. Draw Graph by Hand Range Intervals of increase and decrease Degree: 3 Parent function y = x3 End behavior follows the parent function y = x3 Remember, parent function matches up with the degree. Range, lowest y-value to highest y-value. If necessary, look at the y-values of the min and max to help determine range, and use either [ or ] to include that y-value. (-∞, ∞) Max (-1, 4) x-intercept, y = 0. y-intercept, x = 0 x-intercept 0 = (x-1)2(x+2) (1,0) (-2,0) y-intercept y = (0-1)2(0+2) (0, 2) Look at the x-values of the max and mins to help determine intervals. Use parenthesis. Increase: (- ∞,-1) U (1, ∞) Decrease: (-1, 1) y-int (0, 2) Note, you can have more than one max and min. Also round to hundredths if necessary. x-int (-1, 0) Min\x-int (1, 0) Max (-1, 4) Notice how it touches at x = 1 and crosses at x = -2 Even multiplicity, Touch. Odd multiplicity, Cross Touch at x = 1 Cross at x = -2 Min (1, 0)

Rational Functions\Asymptotes • Domain • Reduce Equation • Intercepts • Even\Odd • Holes & Vertical Asymptotes • Horizontal or Oblique Asymptotes. • Sketch with key points. - (- | – ) (-∞, 1) U (1 , 3) U (3, ∞) Not even (-3, 0) (0, 0) (0, 0) When the degree on bottom is bigger, it is proper, horizontal asymptote is y = 0 When the degree on top is bigger, it is improper, use long division. It will be either horizontal or oblique. REMEMBER PLACEHOLDERS! Don’t forget ‘y =‘ Where the factor cancels out, there is a hole there. To find the y-value of the hole, plug in the x into the reduced equation. After you reduce, the vertical asymptote is where the denominator equals zero. Remember to put ‘x =‘ Domain: Find where the denominator equals zero before you reduce. It is possible you may have to use quadratic formula or have radicals in the denominator. Not odd y – intercept  x = 0 x – intercept  y = 0 f(x) = f(-x) even f(-x) = -f(x) odd Neither

Complex Numbers The important thing to remember is to put these numbers back into standard form: a + bi i2 = -1

Given zero, find other zeros.