1.6 - Solving Polynomial Equations

```1.6 - Solving Polynomial
Equations
MCB4U - Santowski
(A) Review
• To restate the Factor Theorem, if (ax - b) is a factor of P(x),
then P(b/a) = 0.
• A root or a zero is the x value (b/a) that makes the value of
the polynomial zero. They have special graphical
significance as the x-intercepts (i.e. that is the x value when
the function has a value of zero)
• So as an example, if x - 1 is a factor of x3 – 2x2 - 2 + 2, then
P(1) = 0. The other way to state the same idea is that for P(x)
= x3 – 2x2 - 2 + 2, then x - 1 is factor and that x = 1 is root of
P(x) or that one x-intercept of the function is at x = 1.
(B) Rational Root Theorem
• Our previous observations (although limited
in development) led to the following
theorem:
• Given that P(x) = anxn + an-1xn-1 + ….. +
a1x1 + a0, if P(x) = 0 has a rational root of
the form a/b and a/b is in lowest terms, then
a must be a divisor of a0 and b must be a
divisor of an
(C) Rational Root Theorem
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So what does this theorem mean?
If we want to factor the polynomial P(x) = 2x3 – 5x2 + 22x – 10, then we first
need to find a value a/b such that P(a/b) = 0
So the factors of the leading coefficient are {+1,+2} which are then the
possible values for a
The factors of the constant term, -10, are {+1,+2,+5,+10} which are then the
possible values for b
Thus the possible ratios a/b which we can test using the Factor Theorem are
{+1,+½ ,+2,+5/2,+5,+10}
As it then turns out, P(½) turns out to give P(x) = 0, meaning that x – ½ (or 2x
– 1) is a factor of P(x)
From this point on, we can then do the synthetic division (using ½) to find the
quotient and then possibly other factor(s) of P(x)
(C) Rational Root Theorem Example
(D) Examples
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ex.1. Solve 2x3 – 9x2 - 8x = -15
and then show on a GDC
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Now graph both
g(x) = 2x3 – 9x2 - 8x and then
h(x) = -15 and find intersection
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Then graph:
f(x) = 2x3 – 9x2 - 8x + 15
(D) Examples
• Solve 2x3 + 14x - 20 = 9x2 - 5
and then show on a GDC
• Explain that different solution
sets are possible depending on
the number set being used (real
or complex)
(D) Examples
• ex. 3 Solve 2x4 - 3x3 + 2x2 - 6x - 4 = 0 then
graph using roots, points, end behaviour.
Approximate turning points, max/min points, and
intervals of increase and decrease.
• ex 4. The roots of a polynomial are 2, -3, 3 - 2i.
The graph passes through (1, -64). Determine the
equation of the polynomial and sketch.
(E) Examples - Applications
• ex 5. You have a sheet of paper 30 cm long by
20 cm wide. You cut out the 4 corners as squares
and then fold the remaining four sides to make an
open top box.
– (a) Find the equation that represents the formula for the
volume of the box.
– (b) Find the volume if the squares cut out were each 2
cm by 2 cm.
– (c) What are the dimensions of the squares that need to
be removed if the volume is to be 1008 cm3?
(E) Examples - Applications
• The volume of a rectangular-based prism is given
by the formula V(x) = -8x + x3 – 5x2 + 12
– (i) Express the height, width, depth of the prism in
terms of x
– (ii) State any restrictions for x. Justify your choice
– (iii) what would be the dimensions on a box having a
volume of 650 cubic units?
– (iv) now use graphing technology to generate a
reasonable graph for V(x). Justify your window/view
settings
(E) Examples - Applications
• The equation p(m) = 6m5 – 15m4 – 10m3 + 30m2 + 10 relates the
production level, p, in thousands of units as a function of the number
of months of labour since October, m.
• Use graphing technology to graph the function and determine the
following:
– maximums and minimums. Interpret in context
– Intervals of increase and decrease. Interpret
– Explain why it might be realistic to restrict the domain. Explain
and justify a domain restriction
– Would 0<m<3 be a realistic domain restriction?
• Find when the production level is 15,500 units (try this one
algebraically as well)
(E) Examples - Applications
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Use GDC to create a scatter-plot
Use GDC to create and validate
regression eqns
Discuss domain restrictions in each
model
Predict populations in 2006, 2016
What is the best regression model?
Why?
When will the pop. be 35,000,000
According to the quartic and cubic
model, when was the population
less than 25,000,000
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year
Population in ‘000s
1911
7,207
1941
11,507
1961
18,238
1971
21,568
1981
24,820
1986
26,101
1991
28,031
1996
29,672
2001
30,755
• Finding Zeroes of Polynomials from
WTAMU
• Finding Zeroes of Polynomials Tutorial #2
from WTAMU
• Solving Polynomials from Purple Math
(G) Polynomials in Nested Form
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An optional factoring technique that may make it easier for evaluating a
polynomial
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Let P(x) = 2x3 – 3x2 + 5x – 7
Then P(x) = (2x2 – 3x + 5)x – 7
And P(x) = ((2x – 3)x + 5)x – 7
And P(x) = (((2)x – 3)x + 5)x – 7
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So P(4) = (((2)4 – 3)4 + 5)4 – 7
P(4) = ((8 - 3)4 + 5)4 – 7
P(4) = (20 + 5)4 – 7
P(4) = 100 – 7 = 93
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OR P(4) = 2 x 4 = 8 - 3 = 5 x 4 = 20 +5 = 25 x 4 = 100 - 7 = 93
(H) Homework
• Nelson text, page 60, Q1,2,8,9 on the first
day. Graph Q8ac,9ac.
• Nelson text page 61, Q11,12,13,15,19,22,23
on the second day as we focus on
applications of polynomial functions
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