looking for real roots of real polynomials

1. Looking for Real Roots of Real Polynomials
Example E.
a. Let P(x) = 2x3 – 11x2 + 10x + 3, list all the possible
rational roots of P(x).
The factors of 3 are B = {1, 3}.
The factors of 2 are A = {1, 2}
The possible roots are fractions of the form ±b/a
where b is from the set B and a is from the set A.
They are {± , ± , ± , ± }.1
1
3
1
1
2
3
2
b. Factor P(x) into real factors completely.
By trial and error, use synthetic division, we find that
x = 3/2 is a root.
2 –11 10 33/2
2
3
–8
-12
–2
–3
0

2. Looking for Real Roots of Real Polynomials
Hence
2x3 – 11x2 + 10x + 3
= (x – 3/2)(2x2 – 8x – 2)
= (x – 3/2) 2 (x2 – 4x – 1)
= (2x – 3)(x2 – 4x – 1)
x2 – 4x – 1 is an irreducible quadratic polynomial,
by the quadratic formula x = 2 ± 5.
Therefore, P(x) factors completely into real factors:
2x3 – 11x2 + 10x + 3
= (2x – 3)(x – (2 + 5))(x – (2 – 5)).
2 –11 10 33/2
2
3
–8
-12
–2
–3
0

3. Exercise A. (Descartes' Rule of Signs)
Determine the possible number of positive roots and negative
roots of the following polynomials.
Looking for Real Roots of Real Polynomials
B. (Theorem on the Bounds) Gives an interval where the
roots of the following polynomials must reside.
1. P(x) = x3 + x2 + x + 1 2. P(x) = x3 + x2 + x – 1
3. P(x) = x3 + x2 – x + 1 4. P(x) = x3 + x2 – x – 1
5. P(x) = x3 – x2 – x + 1 6. P(x) = x3 – x2 – x – 1
7. What can we say about the roots of a polynomial with
only even degrees of x’s?
8. What can we conclude about the roots of a polynomial
with only odd degrees of x’s?
1. P(x) = x5 + 6x3 + 2x2 – 1 2. P(x) = x4 + 0.01x3 + 0.23x2 – 1/π
3. By the sign-rule, there is at least one positive real root for
P(x) = x4 – 12x3 + 6.8x2 – √101. Graph P(x) using a calculator
over a chosen interval to see if there are more roots.

4. Looking for Real Roots of Real Polynomials
C. (Rational Roots and Factoring Polynomials)
List all the possible rational roots of the following polynomials.
Then find all the rational and irrational roots (all roots are real),
and factor each completely.
1. P(x) = x3 – 2x2 – 5x + 6 2. P(x) = x3 – 3x2 –10x + 6
3. P(x) = –2x3 + 3x2 – 11x – 6 4. P(x) = 3x3 – 4x2 –13x – 6
5. P(x) = –6x3 –13x2 – 4x + 3
6. P(x) = 12x4 – 8x3 – 21x2 + 5x + 6
7. P(x) = 3x4 – x3 – 24x2 – 16x + 8
8. P 𝑥 = 6𝑥4 + 5𝑥3 − 24𝑥2 − 12𝑥 + 16

5. (Answers to the odd problems) Exercise A.
Exercise B.
1. 𝑃(𝑥) has no positive roots and 3 or 1 negative roots.
3. 𝑃(𝑥) has 2 or 0 positive roots and 1 negative root.
5. 𝑃(𝑥) has 2 or 0 positive roots and 1 negative root.
7. It has 0 positive roots and 0 negative roots.
1. 𝑀 =
max{ 1 , 6 , 2 , −1 }
|1|
+ 1 = 7, so the roots reside in (-7,7)
3. 𝑃 𝑥 = 𝑥4– 12𝑥3 + 6.8𝑥2 − 101
The roots are in (-13,13)
Looking for Real Roots of Real Polynomials

6. Exercise C.
1. The possible roots are {±3, ±2, ±1}
𝑃(𝑥) = (𝑥 − 3)(𝑥 − 1)(𝑥 + 2)
3. The possible roots are {±1, ±
1
2
, ±2, ±3, ±6, ±
3
2
}
𝑃(𝑥) = (𝑥 + 3)(𝑥 +
1
2
)(𝑥 − 2)
5. The possible roots are {±1, ±
1
2
, ±
1
3
, ±
1
6
, ±3, ±
3
2
}
𝑃 𝑥 = 2𝑥 + 3 (3𝑥 − 1)(1 − 𝑥)
7. The possible roots are {±1, ±
1
3
, ±2, ±
2
3
, ±4, ±
4
3
, ±8, ±
8
3
}
𝑃 𝑥 = 3𝑥 − 1 𝑥 + 2 (𝑥 + 5 − 1)(𝑥 − 5 − 1)
Looking for Real Roots of Real Polynomials