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Agenda
Thursday, June 23
Technology to approximate roots of polynomial equations
 Relationship between roots/zeros and x-intercepts
 If f(a) is negative and f(b) is positive (or vice versa), the graph must have crossed the xaxis between those points.
 How to find zeros to the nearest tenth on a calculator
 More accurate results with smaller step sizes
 Shortcomings of calculator
 Mentally locate between two consecutive integers a real root of each equation:
 (a) x 3  x  5  0
(b) 2x 3  x  3  0
Solving polynomial equations by factoring
 If roots are nice, can use calculator
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 If #’s are not nice, however, calculator will only give you an approximation. You may
not be able to tell if the roots are rational or irrational.
 Fundamental theorem of Algebra!
 Can only find rational roots for higher degree polynomials (not complex or irrational).
The other roots still exist; you just can’t find them exactly unless you have a linear or a
quadratic function. You can always find an approximation of an irrational root, but the
complex roots do not cross the x-axis and are not visible on the graph
 Can always find all roots for quadratic equations via quadratic formula
 All rational roots will be factors of constant divided by factors of leading coefficient
If an x n  ...  a1 x  a0  p(x) , then roots are factors of ao over factors of an
Board Problem 2
Find all possible rational roots of 3x 3  8x 2  9x  2  0 and then find all roots.
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Board Problem 3
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If possible, find all the zeros of the function f (x)  2x 4  3x 2  20 .
Look for shortcuts in factoring
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Factor like a quadratic: Treat x 2 as if it were x or substitute using y  x 2. Factor like a
quadratic: x 4  3x 2  4  0
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Board Problem 4
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Factor like a quadratic: x 6  x 3  6  0
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Board Problem 5
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Factor by grouping: Find common factors in the first 2 terms and in the second 2 terms. Factor by
grouping: x 3  4 x 2  9x  36  0
Board Problem 6
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Try factoring: x 8  y 8  0
Agenda
Thursday, June 23
General results for polynomial equations
 True/False:
1. Polynomials always have rational coefficients
2. If a polynomial has rational coefficients, the roots have to be rational.
3. If a polynomial has rational coefficients, it has to have an even # of complex roots.
4. If a polynomial has rational coefficients, it has to have an even # of irrational roots.
5. If a polynomial has irrational coefficients, it has to have an even # of irrational roots.
6. If one complex root is 3 – 5i and the coefficients of the polynomial are rational, then
the other root is 3 – 5i.
7. If one root of a polynomial with rational coefficients is 5  2 , then the other root is
5  2.
Board Problem 7
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Find a cubic equation with integral coefficients and roots 1 6 and -3/2 (in factored form – don’t
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have to multiply it out!)
Polynomial function word problems
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Board Problem 8
A shot put champion performs an experiment. He takes different shots, releasing the shot with the
same initial speed but at different angles. The path of each put can be modeled by the following
equations where x is the horizontal distance and y is the vertical distance of the put:
Initial angle of 35 degrees: y  0.010735x 2  0.700208x  6
Initial angle of 65 degrees: y  0.040330x 2  2.144507x  6
(a) Looking at the graphs, what does the x-intercept tell you?
(b) How far does the shot put travel using the different initial angles? Which angle sends the shot
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put a longer distance?
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Board Problem 9
The number of RVs sold in the US from 1985 to 1991 can be modeled by
y  9.5t 2  48.9t  343.5 , where t represents the number of years since 1985 and y the number (in
thousands) of RVs sold.
(a) According to the model, in what year will the number of RVs sold in the US drop to 0?
(b) Is that realistic?