2.8: Polynomial Inequalities And polynomials and graphing and
designing boxes. 2(x 2)(x + 3) F(x)
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Objectives Be able to find the positives, negatives, and zeros
of a polynomial inequality via line method. Be able to graph
polynomial inequalities. Be able to solve inequalities using the
analytical method (synthetic division, rational zeros theorem) Know
what an impossible inequality looks like and understand why its
impossible. Be able to solve rational inequalities. Be able to
combine functions with rational inequalities. Apply this
information to real-life problems (box method, cylinder method,
etc).
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What is an inequality?
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Its not social justice.
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Less Than Or Equal To Greater Than Or Equal To < Less Than
> Greater Than Like Pacman! Symbols:
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Polynomial inequalities! To solve f(x) > 0, find the values
of x that make f(x) positive To solve f(x) < 0, find the values
of x that make f(x) negative To solve f(x) = 0, find the values of
x that make f(x) zero Which can help figure out which part of the
number line is positive or negative. - + -4-405
However some polynomial inequalities do not appear in a
factored form and some dont have zero on one of the sides. But they
can be solved in various ways!
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Solving Them Analytically To do this, use the rational zeros
theorem, finding the possible factors of the constant coefficient
over the leading coefficient. How to find possible factors of a
polynomial, statisticslectures.com
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Examples: Solving a Polynomial Analytically F(x)= 2x-7x-10x+24
In this example, enter the equation on your calculator, and find
the first zero. The next step is synthetic division. 4 2 -7 -10 24
8 4 -24 __________________________________________ 2 1 -6 0 The
first zero occurs at 4
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Continued Continue to factor after youve completed the
synthetic division F(x)= (x-4)(2x + x 6) = (x-4)(2x-3)(x-2) Zeros
are: 4, 3/2, and -2. 1 st zero
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After Using Rational Zeros You may now plug in the possible
rational using synthetic division. After finding a true zero, find
out where the polynomial is positive or negative, as previously
done Synthetic Division Example by
http://taylorlstormberg.wordpress.com
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Graphing! Explained Yet another way to solve the polynomial
inequality is graphically. To do this, make sure that if there are
two polynomials on either side of the inequality get moved to one
side. Then we make the inequality equal to zero. After plugging the
equation into a graphing calculator, find the zeros of the
polynomial. Graphing polynomials from www.algebra-
tutoring.com
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Examples: Solving Graphically x - 6x x-8x Solution: Rewrite!
(-, 0.32] U [1.46, 4.21] x - 6x + 8x 2 0 F(x) = x - 6x + 8x 2 Zeros
approx at: 0.32, 1.46, and 4.21 GRAPH!
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In Addition... Depending on the inequality, the solutions of
the equation may vary. If its x < 0, then the solutions are
below the x-axis If its x > 0, then the solutions are above the
x-axis If its x 0, then the solutions are on or below the x-axis If
its x 0, then the solutions are on or above the x-axis
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However... there are some empty inequalities that have no
solutions
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Impossible Inequalities (x + 7)(2x + 1) < 0(x + 7)(2x + 1) 0
Square roots cannot have negative numbers When solving for zero, x
would equal -7, which would be impossible, unless imaginary numbers
are present. Again, when solving for zero, x would be equal to
either -7 or -1/2. Square roots of negative numbers would still be
impossible unless imaginary numbers are present.
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Examples: Unusual Answers (2x + 7)(- x + 1) 0 solution is (-, )
also, (2x + 7)(- x + 1) 0 solutions is (-, ) BUT, (2x + 7)(- x + 1)
0 (2x + 7)(- x + 1) 0 Both solutions are empty
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Rational Inequalities Along with polynomial inequalities, there
are rational inequalities as well! Not only can the zeros to
rational inequalities be zero, positive, or negative, but zeros can
also be undefined. Example Equation: y = (x+4)(x+3)/(x+-1) -4-4
undefined 3 - + Undefined zeros can be found by finding the zero of
the denominator of the function 0 1 0
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Examples: Sign Charts for a Rational Function R(x) = (2x +1) /
( (x+3)(x-1)) Find the real zeros. ( in numerator only) X = -1/2 If
x is equal to -3 or 1, r(x) is undefined. R(x) is positive if -3 x
-1/2 or x 1. R(x) is negative if x -3 or -1/2 1 1 1 -3 -1/2 (-)
------ (-) ------- (+) (-) (+) ------- (+) (-) (+) ------- (+)
Negative Positive Solution (-3, -1/2) U (1,) and (-, -3) U (-1/2,
1)
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Combining Functions With Rational Inequalities You can also
solve a rational inequality by combining functions. To do this,
combine the two functions on the left-hand side by using the least
common denominator. (multiply the two denominators together). Then
you add the fractions, use distributive property, simplify, and
divide.
Applications Box Problem 1) A packaging company wishes to
design boxes with a volume of at least 600 in. Squares are to be
cut from the corners of a 20-in by 25-in piece of cardboard, with
the flaps folded up to make an open box. What size squares should
be cut? 2) Volume V of a box is V(x) = x(y-2x)(y-2x) 3) Y
represents the side-lengths of cardboard, so Y would go into the
volume equation. 4) X represents the side length of the removed
squares and the height of the box. Volume is now V(x) =
x(25-2x)(20-2x) 5) To obtain a volume of at least 600 in, we make
the equation into an inequality. 600 x(25-2x)(20-2x) 6) To solve
the application graphically, plug the equation from step four into
the calculator and plug 600 into the second y= spot below the first
equation. 7) Make the window 0 x 10 (because the width of the
cardboard is 20) and 0y1000 8) Find the intercepts between the two
graphs. The solution is [1.66, 6.16]
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Applications Can Problem 1. A cannery wants to package cans in
a 2-liter(2000cm) cylindrical can. Find the radius and height of
the cans if the cans have a surface area that is less than 1000 cm.
2. Surface Area = 2 r + 4000/r 3. The inequality for this equation
is 2 r + 4000/r < 1000 4. To solve this equation plug the
equation from step two into the graphing calculator. Then put 1000
in the second y= area below that. 5. Find out where the lines cross
via graphing. The surface area should be 4.62 < r < 9.65 to
be less than 1000 cm. 6. The volume equation is V = rh and V =
2000. 7. Using this equation, we can find out that H = 2000/ r 8.
To find values of h we build a double inequality for 2000/( r)