Download Arvind Borde MATH 1: Week 9 Polynomial Equations and Graphs

Arvind Borde
MATH 1: Week 9
Polynomial Equations and Graphs
A monomial in one variable, x, is an
expression of the type axn , where a is any
number and n is a non-negative integer.
Examples are
1
2
5x ,
2 3
− x ,
3
7,
and πx756 .
Observe that 7 (like any fixed number) is
considered a monomial because
7 = 7(1) = 7x0 .
The following are not monomials
(1) 3x−2
√
(2) x
(3) 1/x
In each case, why not?
2
The following are not monomials
(1) 3x−2
√
(2) x
(3) 1/x
In each case, why not?
Answer(s)
(1) Negative power.
√
(2) x = x1/2 . Fractional power.
(3) 1/x = x−1 . Negative power.
3
A polynomial in one variable x is an
expression that consists of adding and
subtracting monomials in x.
Which of these is a polynomial?
(4) x3 − (1/2)x + 3 .
(5) x2 + 1/x .
√
17
3
(6) x − 5x + x + x2 − 35.54 .
A polynomial in one variable x is an
expression that consists of adding and
subtracting monomials in x.
Which of these is a polynomial?
(4) x3 − (1/2)x + 3 .
(5) x2 + 1/x .
√
17
3
(6) x − 5x + x + x2 − 35.54 .
Answer(s)
(4) Polynomial.
(5) Not polynomial (because of 1/x).
√
(6) Not polynomial (because of x).
We classify polynomials by their degree: the
highest power of the variable.
For example 2x3 − 3x2 + x − 6 has degree 3.
4
What is the degree of
(7) 3x4 − x + 1?
(8) −2x5 − x4 + x3 − 2x6 + x2 + 4?
We classify polynomials by their degree: the
highest power of the variable.
For example 2x3 − 3x2 + x − 6 has degree 3.
What is the degree of
(7) 3x4 − x + 1?
(8) −2x5 − x4 + x3 − 2x6 + x2 + 4?
Answer(s)
(7) 4.
(8) 6.
We are interested in the zeros of a
polynomial, and in its graph.
A zero of a polynomial is a value of x that
makes the value of the polynomial zero.
5
For example, x = 1 is a zero of x2 − 1.
(Plug in x = 1: 12 − 1 = 1 − 1 = 0.)
We’ll study polynomials in order of increasing
degree.
Degree 0: Constant Polynomials
Examples:
7x0 (= 7).
−(12/5)x0 (= −(12/5)).
6
Polynomials of degree zero are simply
numbers, and there’s nothing new to discuss
here.
Degree 1: Linear Polynomials
Examples:
−5x + 1
(4/2)x + π
7
To get the zeros of the first of these:
−5x + 1 = 0
1 = 5x
x = 1/5
(9) What are the zeros of 3x − 15?
8
(9) What are the zeros of 3x − 15?
3x − 15 = 0
+15
3x
+15
= 15
3x
15
=
3
3
x=5
Getting zeros is basically solving equations.
(10) Solve 4x − 2 = 13 − 2x
9
(10) Solve 4x − 2 = 13 − 2x
4x − 2 = 13 − 2x
+2
= 15 − 2x
4x
+2x
6x
+2
+2x
= 15
6x
15
5 ×6 3
=
=
6
6
2 ×6 3
x = 5/2
Graphs of Linear Polynomials
To plot the linear polynomial 2x − 1:
1) Write the equation y = 2x − 1.
2) Make a table:
10
x
( 0
Chosen
1
2
2x − 1
2(0) − 1
2(1) − 1
2(2) − 1
y
−1 )
Calculated
1
3
3) Plot the (x, y) values and connect them.
Here’s the graph of y = 2x − 1:
11
y-int
→
- x-int
Note:
1) y-intercept is −1. Why?
2) x-intercept is the solution of 0 = 2x − 1;
i.e., the zero of 2x − 1: x = 1/2.
3) Graph points upward.
(11) Plot −2x + 3 and get the intercepts.
12
x −2x + 3
y
0 −2(0) + 3 3
1 −2(1) + 3 1
2 −2(2) + 3 −1
Note:
1) y-intercept is 3.
2) x-intercept (solution of 0 = −2x + 3): 3/2.
3) Graph points downward.
The General Linear Polynomial
ax + b
13
1) Graph points upward (reading from left
to right) when a > 0, and downward when
a < 0. (a is called the slope.)
2) The y-intercept is b.
3) The x-intercept is the zero of ax + b.
Degree 2: Quadratic Polynomials
Examples:
−5x2 + x + 1
(4/2)x + x2
14
To get the zeros of the first of these you must
solve the quadratic equation
−5x2 + x + 1 = 0.
Solving Quadratic Equations
The general quadratic equation is
ax2 + bx + c = 0
15
where a, b, and c are fixed numbers.
The solution can always be obtained from the
quadratic formula:
q
−b ±
b2 − 4ac
x=
.
2a
Example: What are the zeros of x2 − 5x + 6?
16
To answer this you must solve x2 −5x+6 = 0.
Here a = 1, b = −5, and c = 6. Therefore,
p
−(−5) ± (−5)2 − 4(1)(6)
x=
2(1)
√
√
5± 1
5 ± 25 − 24
=
=
2 
2
(5 + 1)/2 = 6/2 = 3
5±1 
=
=
or

2
(5 − 1)/2 = 4/2 = 2
(12) What are the zeros of 3x2 − 2x − 1?
17
(12) What are the zeros of 3x2 − 2x − 1?
a = 3, b = −2, and c = −1. Therefore,
p
−(−2) ± (−2)2 − 4(3)(−1)
x=
2(3)
√
√
2 ± 16
2 ± 4 + 12
=
=
6 
6
(2 + 4)/6 = 6/6 = 1
2±4 
=
=
or

6
(2 − 4)/6 = −2/6 = −1/3
(13) What are the zeros of 2x2 + 3x − 1?
18
(13) What are the zeros of 2x2 + 3x − 1?
a = 2, b = 3, and c = −1. Therefore,
x=
=
√
−3 ±
−3 ±
p
√
4
32 − 4(2)(−1)
2(2)
9+8
√
−3 ± 17
=
4
Since 17 has no simpler exact value, you can
leave the answer as it is.
(14) What are the zeros of x2 − 2x + 1?
19
(14) What are the zeros of x2 − 2x + 1?
a = 1, b = −2, and c = 1. Therefore,
p
(−2)2 − 4(1)(1)
x=
2(1)
√
√
2± 4−4
2± 0
=
=
2
2
2±0
2
=
= = 1.
2
2
−(−2) ±
In this case there is a single solution.
(15) What are the zeros of x2 + 1?
20
(15) What are the zeros of x2 + 1?
a = 1, b = 0, and c = 1. Therefore,
p
(0)2 − 4(1)(1)
x=
2(1)
√
√
0 ± −4
± −4
=
=
.
2
2
−(0) ±
√
−4 is not a real number, so in this case we
say there is no real solution.
(16) Can you give a verbal argument why no
real values of x can make x2 + 1 = 0?
21
(16) Can you give a verbal argument why no
real values of x can make x2 + 1 = 0?
If x2 + 1 = 0, then x2 = −1. But x2 must be
non-negative.
The General Quadratic Polynomial
ax2 + bx + c
1) Graph points upward when a > 0, and
downward when a < 0.
22
2) The y-intercept is c (the pure number).
3) The x-intercepts are zeros of ax2 + bx + c.
23
(17) We have seen that a quadratic
expression can have two, one, or no zeros.
This must match situations where the
associated graph has two, one, or no xintercepts. Sketch all three possibilities.
(17) We have seen that a quadratic
expression can have two, one, or no zeros.
This must match situations where the
associated graph has two, one, or no xintercepts. Sketch all three possibilities.
Important emerging moral: The zeros of a
polynomial are the x-intercepts of its graph.
The General Cubic Polynomial
ax3 + bx2 + cx + d
1) Graph points ↓down-up↑ when a > 0, and
↑up-down↓ when a < 0.
24
2) The y-intercept is d (the pure number).
3) The x-intercepts are zeros of the
polynomial.
↑
↑
↓
↓
You can get the roots of ax3 + bx2 + cx + d
either graphically, by reading off the xintercepts of the graph, or from the cubic
formula – too complicated to show you here.
25
Just kidding . . .
The General Quartic Polynomial
ax4 + bx3 + cx2 + dx + e
1) Graph points ↑up-up↑ when a > 0, and
↓down-down↓ when a < 0.
26
2) The y-intercept is e (the pure number).
3) The x-intercepts are zeros of the
polynomial.
↑
↑
↓
↓
You can get the roots of
ax4 + bx3 + cx2 + dx + e
27
either graphically, by reading off the xintercepts of the graph, or from the quartic
formula – too complicated to show you here.
I kid you not.
28
(18) We saw (question 17) that a quadratic
could have two, one, or zero x-intercepts
(and, therefore, zeros). We see that the
quartics on the previous page have four xintercepts. Can you sketch quartics with
three, two, one and zero x-intercepts?
(Or, am I just messing with you?)
(18) We saw (question 17) that a quadratic
could have two, one, or zero x-intercepts
(and, therefore, zeros). We see that the
quartics on the previous page have four xintercepts. Can you sketch quartics with
three, two, one and zero x-intercepts?
(Or, am I just messing with you?)
(I respect you too much to mess with you.)
(19) We see that the cubics on page 24 have
three x-intercepts. Can you sketch cubics with
two, one and zero x-intercepts?
(Or, am I just messing with you?)
29
(19) We see that the cubics on page 24 have
three x-intercepts. Can you sketch cubics with
two, one and zero x-intercepts?
(Or, am I just messing with you?)
Just messing.
The General Quintic Polynomial
ax5 + bx4 + cx3 + dx2 + ex + f
(20) Choosing from
↑up-up↑, ↑up-down↓, ↓down-up↑, ↓down-down↓,
which way will the graph point when
a > 0 and when a < 0?
30
(21) What is the y-intercept?
(22) How would you get the x-intercepts?
What are all the possibilities (number) that
you expect?
31
Answer(s)
(20) ↓down-up↑ when a > 0, and ↑up-down↓
when a < 0.
(21) f
(22) Zeros of the polynomial.
Between one and five x-intercepts.
↑
↑
↓
↓
Zeros of Quintics (and higher degrees)
32
Starting with quintic polynomials and going to
higher degrees we have no formulas (involving
finite numbers of addition, subtractions,
divisions, multiplications and roots of the
coefficients) that give us the roots.
Graphical methods can always be used.
Summary of polynomials of any degree n
axn + lower powers +N
n even, a > 0: Graph points ↑up-up↑.
n even, a < 0: Graph points ↓down-down↓
33
n odd, a > 0: Graph points ↓down-up↑.
n odd, a < 0: Graph points ↑up-down↓.
In every case, N is the y-intercept.
In every case, the x-intercepts are the zeros of
the polynomial.