Transcript Section 5.1 - Gordon State College

Section 5.1
Solving Polynomial Equations
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POLYNOMIAL EQUATIONS
In Chapter 1, we solved linear (first-degree)
polynomial equations. In Chapter 2, we solved
quadratic (second-degree) polynomial equations.
In Chapter 2, we also mentioned polynomials
with higher-degree. In this section, we want to
learn to solve higher-degree polynomials. We
also want to see how polynomials of higherdegree can be applied to real-world situations
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CUBIC POLYNOMIALS
A cubic (third-degree) polynomial equation has
the form
ax3 + bx2 + cx + d = 0.
A cubic polynomial can have as many as three
real number solutions.
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EXAMPLES OF CUBIC
POLYNOMIALS
1. y  x  3 x  1
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2. y  x  3 x  2
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3. y  x  3 x  6
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POPCORN TRAY PROBLEM
You are designing open-topped popcorn trays out of
rectangular pieces of cardboard. Each tray will be
constructed by cutting equal squares out of the corners,
and then folding the remaining flaps up to form a box.
(See the diagram.) Your task is to investigate the volume
of the popcorn tray.
We see the volume is
given by
V = LWx.
Or, V = x(p − 2x) (q − 2x)
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EXAMPLE
Starting with a 15 cm by 35 cm sheet of
cardboard, a popcorn tray is constructed. If the
volume is to be precisely 450 cm3, what are the
dimensions of the popcorn tray?
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QUARTIC POLYNOMIALS
A quartic (fourth-degree) polynomial equation
has the form
ax4 + bx3 + cx2 + dx + e = 0.
A quartic equation can have as many as four real
number solutions.
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EXAMPLES OF QUARTIC
POLYNOMIALS
1. y  x  4 x  9 x  3
4
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2. y  x  4 x  9 x  2
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A quartic polynomial has
• three or fewer bends (turning points)
• four or fewer real-number zeros.
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EXAMPLE
A 25-foot ladder is leaning
across an 8-foot fence and
just touches a tall wall
standing 5 feet behind the
fence (see diagram). How
far (accurate to the nearest
tenth of an inch) is the
base of the ladder from the
bottom of the fence?
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