4 ESO Academics - Unit 03 - Exercises 4.3.4 - The Remainder Theorem. Roots of a Polynomial.

1. UNIT 03 December
1. Find the remainder of these divisions using Ruffini´s rule and the remainder
theorem. Check that Check that the results match
a) (𝒙𝒙𝟐𝟐
+ 𝟑𝟑𝒙𝒙 − 𝟒𝟒) ÷ (𝒙𝒙 − 𝟓𝟓)
b) (𝟐𝟐𝒙𝒙𝟑𝟑
− 𝟓𝟓𝒙𝒙 + 𝟕𝟕) ÷ (𝒙𝒙 − 𝟐𝟐)
c) (−𝒙𝒙𝟑𝟑
+ 𝟐𝟐𝒙𝒙 − 𝟏𝟏) ÷ (𝒙𝒙 + 𝟏𝟏)
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.4.1

2. UNIT 03 December
2. Without doing the division write the remainder of:
(𝒙𝒙𝟑𝟑
+ 𝟑𝟑𝟑𝟑 − 𝟒𝟒) ÷ (𝒙𝒙 − 𝟏𝟏)
3. Find the coefficient m of polynomial 𝑷𝑷(𝒙𝒙) = 𝒙𝒙𝟑𝟑
+ 𝒎𝒎𝒎𝒎 − 𝟑𝟑 knowing that the
remainder of 𝑷𝑷(𝒙𝒙) ÷ (𝒙𝒙 − 𝟐𝟐) is 5.
4. Decide if the following numbers are roots of the polynomial
𝑷𝑷(𝒙𝒙) = 𝒙𝒙𝟑𝟑
− 𝟐𝟐𝒙𝒙𝟐𝟐
− 𝐱𝐱 + 𝟐𝟐
a) 𝒙𝒙 = 𝟐𝟐
b) 𝒙𝒙 = −𝟏𝟏
c) 𝒙𝒙 = 𝟏𝟏
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.4.2

3. UNIT 03 December
d) 𝒙𝒙 = −𝟓𝟓
5. Calculate the integer roots of 𝑷𝑷(𝒙𝒙) = 𝒙𝒙𝟑𝟑
+ 𝒙𝒙𝟐𝟐
− 𝟐𝟐𝟐𝟐 − 𝟐𝟐
6. Write two polynomials that have 𝒙𝒙 = 𝟏𝟏 as root.
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.4.3