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Analysis BC
Miss Hall
Unit 5 Day 4: Irrational Zeros & IVT
Complex Root Theorem: Given a polynomial function, f, if a + bi is a root of the
polynomial then a – bi must also be a root.
Example: Find a polynomial with rational coefficients with zeros 2, 1 +
3 , and 1 – i.
Intermediate Value Theorem (IVT): Given real numbers a & b where a < b. If a
polynomial function, f, is such that f(a) ≠ f(b) then in the interval [a, b] f takes on every
value between f(a) to f(b).
Examples:
1) First use your calculator to find the zeros of f ( x)  3 x 4  x3  2 x 2  5 .
8
Now verify the1 unit integral interval that the zeros are in using the Intermediate
Value Theorem.
2) Use the Intermediate Value Theorem to find the 1 unit integral interval for each of
the indicated number of zeros.
a. One zero: g ( x)  3x3  4 x 2  x  3
b. Four zeros: f ( x)  x 4  10 x 2  2
Analysis BC
Miss Hall
Unit 5 Day 4: Irrational Zeros & IVT
3) Given f ( x)  3x3  4 x 2  3x  2 :
a. What is a value guaranteed to be between f(2) and f(3).
b. What is another value guaranteed to be there?
c. What is a value that is NOT guaranteed to be there?
d. But could your value for c be there? Sketch a graph to demonstrate your
answer.
4) Given a polynomial, g, where g(0) = -5 and g(3) = 15:
a. True or False: There must be at least one zero to the polynomial. Explain.
b. True or False: There must be an x value between 0 and 3 such that g(x) = 12.
Explain.
c. True or False: There can not be a value, c, between 0 and 3 such that g(c) = 25.
Explain.
Homework:
1) Find the equation of the quadratic whose only x-intercept is (-4, 0) and passes through
(-2, 8)
Analysis BC
Miss Hall
Unit 5 Day 4: Irrational Zeros & IVT
2) Graph y = -½ x2 + ½ x + 3.
3) Given f ( x)  x6  6 x5  8 x 4  14 x3  45 x 2  40 x  12
a. What is the end behavior
b. List the possible rational roots
4) Divide: (2 x5  8 x3  2 x 2  4 x  2)  (2 x  4)
5) Use the remainder theorem to find f(5) for f(x) from problem #3.
6) Use your calculator to approximate the real zeros and relative extrema of the following
functions.
a. f ( x)  2 x 4  6 x 2  1
b.
f ( x)  x 5  3 x 3  x  6
Analysis BC
Miss Hall
Unit 5 Day 4: Irrational Zeros & IVT
7) Find a polynomial function that has the given zeros.
a. 0, 4
b. 0, -2, -3
c. 4, -3, 3, 0
d. 1 +
3
e. 2, 4 – 5
8) For each of the following: (a) use the IVT to find integral intervals one in length which
must contain a zero (b) now use your calculator to find the zeros (checking your answer
to part (a).
a. f ( x)  x3  3x 2  3
b. g ( x)  3x 4  4 x3  3
Analysis BC
Miss Hall
Unit 5 Day 4: Irrational Zeros & IVT
9) For each of the following: Identify the symmetry it has (x-axis, y-axis or origin), and
determine the number of x-intercepts it has. Use your calculator to verify your answer.
a. f ( x)  x 2 ( x  6)
1
b. g (t )  (t  4) 2 (t  4) 2
2
c.
f ( x)  x 3  4 x
d.
1
f ( x)  ( x  1) 2 ( x  3)(2 x  9)
5
10) An open box is to be made from a square piece of material 36cm on a side by cutting
equal squares with sides of length x from the corners and turning up the sides.
a. Find an equation for the volume of the box, V(x)
b. Determine the domain of the function V
c. Use your calculator to find the length, x, for which the maximum volume is
produced.
Analysis BC
Miss Hall
Unit 5 Day 4: Irrational Zeros & IVT
Answers: 1) y = 2x2 + 16x + 32 2) check with calc
63
4) x  2 x  8 x  15 x  32 
x2
4
3
2
3) a) up, up b) 12, 6, 4, 3, 2, 1
zeros :  1.680, 0.421
5) 2688 6) a) Re l Max :(0,1)
Abs Min : (1.225, 3.500)
zeros :  1.178
b) Re l Max :(0.324, 6.218)
Re l Min : (0.324,5.782)
7) a) f ( x)  x 2  4 x b) f ( x)  x3  5x 2  6 x
c) f ( x)  x 4  4 x3  9 x 2  36 x
d) f ( x)  x 2  2 x  2 e) f ( x)  x3  10 x 2  27 x  22
8) a) (-1,0) (1, 2) (2, 3): -0.879, 1.347, 2.532
9) check with calc
b) (-2, -1) (0,1): -1.585, 0.779
10) a) V(x) = x(36 – 2x)2 b) Domain (0, 18) c) 6cm