13 graphs of factorable polynomials x

Graphs of Factorable Polynomials

A polynomial P(x) = anxn + an-1xn-1 + .. + a1x + a0
is said to be factorable if (using only real numbers)
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk

P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2

P(x) = an(x – r1) (x – r2) .. (x – rk) .
Hence r1, r2,.. ,rk are the roots of P(x).
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2

P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
So P(x) is factorable with roots x = 0, –2, and 2.

P(x) = an(x – r1) (x – r2) .. (x – rk) .
The order of or the power of a root is the
corresponding power raised in the factored form, i.e.
the order of the r1 is N1, order of r2 is N2 ,etc..
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2

P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
x = 0 has order 3, x = –2 and x = 2 have order 2.

Behaviors of factorable polynomials give us insights to
behaviors of all polynomials.
P(x) = an(x – r1) (x – r2) .. (x – rk) .
N1 N2 Nk
Example A: P(x) = 2x7 – 16x5 + 32x3
= 2x3(x4 – 8x2 + 16) = 2x3(x2 – 4)2
= 2(x – 0)3(x + 2)2(x – 2)2
x = 0 has order 3, x = –2 and x = 2 have order 2.

In this and the next sections, we devise a strategy for
graphing factorable polynomial and rational functions.

A basic fact about the graphs of polynomials is that
their graphs are continuous lines:
x
x
Graphs of Polynomials

x
x
x x
Graphs of Non–polynomials

x
x
x x
To graph a function, we separate the job into two parts.

x
x
x x
I. What does the graph look like in the “middle”,
i.e. the curvy portion that we draw on papers?

x
x
x x
II. What does the graph look like beyond the drawn
area?

x
x
x x
II. What does the graph look like beyond the drawn
area?
We start with part II with factorable polynomials.

We start with the graphs of the polynomials y = ±xN.

The graphs y = xeven
y = x2

y = x2
y = x4
(1, 1)
(-1, 1)

y = x2
y = x4
y = x6
(1, 1)
(-1, 1)

y = x2
y = x4
y = x6
y = -x2
y = -x4
y = -x6
(1, 1)
(-1, 1)
(-1,-1) (1,-1)

y = x2
y = x4
y = x6
y = -x2
y = -x4
y = -x6
y = ±xeven:
y = xeven y = –xeven
(1, 1)
(-1, 1)
(-1,-1) (1,-1)

y = x3
The graphs y = xodd

y = x3
y = x5
(1, 1)
(-1, -1)
The graphs y = xodd

y = x3
y = x5
y = x7
(1, 1)
(-1, -1)
The graphs y = xodd

The graphs y = xodd
y = x3
y = x5
y = x7 y = -x3
y = -x5
y = -x7
(1, 1)
(-1, -1)
(-1, 1)
(1,-1)

The graphs y = xodd
y = x3
y = x5
y = x7 y = -x3
y = -x5
y = -x7
y = ±xodd
y = xodd y = –xodd
(1, 1)
(-1, -1)
(-1, 1)
(1,-1)

y = ±xodd
y = ±xeven:

Facts about the graphs of polynomials:
x
x

• The graphs of polynomials are unbroken curves.
x
x

x x
(broken or discontinuous)
x
x

• Polynomial curves are smooth (no corners).
x
x
x x

• Polynomial curves are smooth (no corners).
x
x
x x
(not smooth, has corners)
x

Let P(x) = Axn (Head) + lower degree terms (Tail).

For large |x|, the Head Axn (the leading term)
dominates the Tail (lower degree terms).

That is, for x's to the far left and right (| x |'s are large),
the Tail is negligible compare to Axn.

Let’s take P(x) = x5 (Head) + 1000 x4 (Tail) as an example.
The ratio of the Tail: Head is 1000 x4/ x5 =1000/x.

For “large” x's (say x = 10100, one google), 1000/x ≈ 0.
This means the Tail 1000x4 contributes a negligible amount compared to the
Head x5. So y = x5 shapes the graph of y = P(x) = x5 + 1000 x4 for large |x|'s.

Hence, for x where |x| is "large",
y = P(x) resembles y = Axn (the Head)

Realizing this, we see there’re four types of graphs
of polynomials, to the far left or far right, basing on
i. the sign of the leading term Axn and

Realizing this, we see there’re four types of graphs
of polynomials, to the far left or far right, basing on
i. the sign of the leading term Axn and
ii. whether n is even or odd.

Behaviors of polynomial-graphs to the "sides":

y = +xeven + lower degree terms:

y = +xeven + lower degree terms: y = –xeven + lower degree terms:

y = +xodd + lower degree terms:

y = +xodd + lower degree terms: y = –xodd + lower degree terms:

For factorable polynomials, we use the sign-charts to
sketch the central portion of the graphs.

Recall that given a polynomial P(x), it's sign-chart is
constructed in the following manner:

Construction of the sign-chart of polynomial P(x):

I. Find the roots of P(x) and their order respectively.

II. Draw the real line, mark off the answers from I.

III. Sample a point for it's sign, use the orders of the
roots to extend and fill in the signs.

III. Sample a point for it's sign, use the orders of the
roots to extend and fill in the signs.
Reminder:
Across an odd-ordered root, sign changes
Across an even-ordered root, sign stays the same.

Example B. Make the sign-chart of f(x) = x2 – 3x – 4

Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0

Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0
The roots are x = 4 , -1 and both are odd-ordered.

Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0
Mark off these points on a line.

Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0
Mark off these points on a line.
4
-1

Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0
Mark off these points on a line. Test for sign using x
= 0 and we get f(0) negative.
4
-1

Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0
= 0 and we get f(0) negative.
0 4
-1
Test x = 0,
we get that
f(0) negative.

Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0
= 0 and we get f(0) negative. Since both roots are
odd-ordered, the sign changes to "+" across the
them.
0 4
-1
Test x = 0,
we get that
f(0) negative.

Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0
them.
0 4
-1
Test x = 0,
we get that
f(0) negative.
+ +

Solve x2 – 3x – 4 = 0
(x – 4)(x + 1) = 0
them.
0 4
-1
Test x = 0,
we get that
f(0) negative.
The graph of y = x2 – 3x – 4 is shown below:
+ +

+ + + + + – – – – – + + + + +
4
-1
y=(x – 4)(x+1)

Note the sign-chart reflects the graph:
I. The graph touches or crosses the x-axis at the roots.
+ + + + + – – – – – + + + + +
4
-1
y=(x – 4)(x+1)

II. The graph is above the x-axis where the sign is "+".
+ + + + + – – – – – + + + + +
4
-1
y=(x – 4)(x+1)

II. The graph is above the x-axis where the sign is "+".
III. The graph is below the x-axis where the sign is "–".
+ + + + + – – – – – + + + + +
4
-1
y=(x – 4)(x+1)

II. The “Mid-Portions” of Polynomial Graphs

y = ±xodd
y = ±xeven:
y = x y = –x

Graphs of an odd ordered root (x – r)odd at x = r.

+ +
order = 1
r r
y = (x – r)1
y = –(x – r)1

+ +
order = 1
r r
order = 3, 5, 7..
y = (x – r)1
y = –(x – r)1
+
r
+
r
y = (x – r)3 or 5.. y = –(x – r)3 or 5..

order = 2, 4, 6 ..
+
+
x=r
r
Graphs of an even ordered root at (x – r)even at x = r.
order = 2, 4, 6 ..
y = (x – r)2 or 4.. y = –(x – r)2 or 4..

order = 2, 4, 6 ..
+
+
If we know the roots of a factorable polynomial,
then we may construct the central portion of the
graph (the body) in the following manner using its
sign chart.
x=r
r
order = 2, 4, 6 ..
y = (x – r)2 or 4.. y = –(x – r)2 or 4..

order = 2, 4, 6 ..
+
+
If we know the roots of a factorable polynomial,
then we may construct the central portion of the
graph (the body) in the following manner using its
sign chart.
I. Draw the graph about each root using the
information about the order of each root.
II. Connect all the pieces together to form the graph.
x=r
r
order = 2, 4, 6 ..
y = (x – r)2 or 4.. y = –(x – r)2 or 4..

For example, given two
roots with their orders and
the sign-chart of a
polynomial,

the sign-chart of a
polynomial,
+
order=2 order=3

the sign-chart of a
polynomial, the graphs
around each root are as
shown.
+
order=2 order=3

the sign-chart of a
shown. Connect them to get
the whole graph.
+
order=2 order=3

the sign-chart of a
the whole graph.
Example C. Given P(x) = -x(x + 2)2(x – 3)2, identify
the roots and their orders. Make the sign-chart.
Sketch the graph about each root. Connect them to
complete the graph.
+
order=2 order=3

the sign-chart of a
the whole graph.
complete the graph.
The roots are x = 0 of order 1,
+
order=2 order=3

the sign-chart of a
the whole graph.
complete the graph.
The roots are x = 0 of order 1, x = -2 of order 2,
and x = 3 of order 2.
+
order=2 order=3

The sign-chart of P(x) = -x(x + 2)2(x – 3)2 is
+
+
x = 3
order 2
x = 0
order 1
x = -2
order 2

+
+
x = 3
order 2
x = 0
order 1
x = -2
order 2
By the sign-chart and the order of each root, we draw
the graph about each root.

+
+
x = 3
order 2
x = 0
order 1
x = -2
order 2
the graph about each root. (Note for x = 0 of order 1,
the graph approximates a line going through the point.)

+
+
x = 3
order 2
x = 0
order 1
x = -2
order 2
Connect all the pieces to get the graph of P(x).

x = -2
order 2
+
+
x = 0
order 1
x = 3
order 2
Connect all the pieces to get the graph of P(x).

Note the graph resembles its leading term: y = –x5,
when viewed at a distance:

Note the graph resembles its leading term: y = –x5,
when viewed at a distance:
-2
+
+
0 3

Graphs of Rational Functions
The graphs around roots (zero).
Odd-order Roots
+
+ +
order = 1 order = 1
+
order = 3, 5, 7..
+
+
Even-order Roots

Exercise A. Following are the sign charts of
factorable polynomials with their roots and orders.
a. Write down a (any) polynomial in the factored form of
any given sign chart. b. Sketch its graphs.
–1
1.
ord=2 ord=1
1
– – – –1
2.
ord=2
ord=1
1
+ + +
–1
3.
ord= 2 ord= 2
1 – – – –1
4.
ord=1
ord= 3
1
+ +
–1
5.
ord=2 ord=1
1 3
ord=1
–1
6.
ord=3 ord=2
1 3
ord=1
–1
7.
ord=2 ord=2
1 3
ord=3
–1
8.
ord=3 ord=2
1 3
ord=2
–1
9.
ord=2 ord=3
1 3
ord=2
–1
10.
ord=3 ord=2
1 3
ord=3
+ +
+ +
+ + – –
– –
– –

B. For the following polynomials, identify the roots and
their orders. Make sign charts then sketch the graphs.
1. P(x) = (x – 3)2
2. P(x) = –x(x + 2)
3. P(x) = –x(x – 3)2
4. P(x) = x2(x + 2)
5. P(x) = –x2(x – 3)2 6. P(x) = –x(x + 2)2(x – 3)
7. P(x) = x2(x + 2)(x – 3)2 8. P(x) = –x(x + 2)2(x – 3)2
9. P(x) = –x3(x + 2)2(x – 3) 10. P(x) = x2(x + 2)2(x – 3)2
11. P(x) = x2(x + 2)2(x – 3) 12. P(x) = –x(x + 2)4(x – 3)3
13. P(x) = x2(x + 2)2(x – 3)(5 – x)2
14. P(x) = x(x + 2)3(x – 3)2(5 – x)3

(Answers to odd problems) Exercise A.
1. (x + 1)2(x – 1) 3. – (x + 1)2(x – 1) 2 5. – (x + 1)²(x – 1) (x – 3)
7. (x + 1)2(x – 1)2(x – 3)3 9. –(x + 1)2(x – 1)3(x – 3)2

Exercise B.
1.
ord=2
3
+ + + + + + 3.
ord=1
0
+ + – –
ord=2
3 – – 5.
ord=2
0 – –
ord=2
3 – –
– –
7.
ord=2
0
ord=2
3
– –
ord=1
– 2 + + + + + + 9.
ord=3
0
ord=1
3
– –
ord=2
–2 + +
– – – –

11.
ord=2
0
ord=1
3
– –
ord=2
–2 + +
– – – –
13.
ord=2
0
ord=1
3
– –
ord=2
–2 + +
– – – –
ord=2
5

13 graphs of factorable polynomials x

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13 graphs of factorable polynomials x