Chapter 13
Roots of Polynomials
Recap Part 2
1
Contents
• Remainder and Factor Theorem
• Fundamental Theorem of Algebra
• Rational Roots
• Upper and lower bounds
• Descartes rule of signs
• Conjugate pairs
2
Theorems
• Remainder Theorem: If f(x) is divided by (x-r) then the
remainder is f(r)
• Factor Theorem: If the remainder is zero, then r is a root
3
Rational Roots Theorem
• For the polynomial anxn + an-1xn-1 + …a1x + a0 = 0,
n≥1 and an≠ 0, and the coefficients are integers
If p/q is a rational root of the equation, and p and q have no
common factors other than ± 1 (relatively prime)
Then p is a factor of a0 and q is a factor of an.
4
Upper and Lower Bounds
• If, when performing synthetic division of a polynomial with real
coefficients by a potential positive root, the answer has all
positive signs, then the potential root is an upper bound for all
rational roots
• If, when performing synthetic division with a potential negative
root, the answer has alternating signs, then the potential root
is a lower bound for all rational roots
• These two rules can eliminate the number of rational roots
you need to try out!
5
Descartes Rule of Signs
• In a polynomial, with real coefficients the number of positive
roots is equal to the number of sign changes, or less than that
by an even number
• If one substitutes –x for x in the polynomial f(x), the number of
negative roots is equal to the number sign changes in the
resulting polynomial or less than that by an even number
6
Conjugate Roots
• For equations with rational coefficients:
– Complex roots come in pairs: a + bi and a – bi
– Roots of the form a + 𝑏 also come in pairs:
if a + 𝑏 is a root, so is a - 𝑏
7
Example
• Find the integers hat are upper and lower bounds for the real
roots of the following:
• x3 + 2x2 – 5x + 20 = 0
• x5 – 3x2 + 100 = 0
• 2x4 – 7x3 – 5x2 + 28x – 12 = 0
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Solution
• x3 + 2x2 – 5x + 20 = 0
• 2 is upper bound, -5 is lower bound
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Solution
• x5 – 3x2 + 100 = 0
• 2 is upper bound, -4 is lower bound
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Solution
• 2x4 – 7x3 – 5x2 + 28x – 12 = 0
• 6 is upper bound, -2 is lower bound
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Example
• Investigate the roots of 2x4 – 3x3 + 12x2 + 22x -60
12
Solution
• Investigate the roots of 2x4 – 3x3 + 12x2 + 22x -60
• One negative root
• One or three positive roots
13
Example
• Investigate the roots of: x3 + 8x + 5
14
Solution
• Investigate the roots of: x3 + 8x + 5
• No positive roots
• One negative root
• Must have two complex roots
15