The document contains math exercises involving dividing polynomials using Ruffini's rule and the remainder theorem to find the remainders, identifying coefficients based on known remainders, determining if numbers are roots of polynomials, finding integer roots of polynomials, and writing polynomials with a given root. Exercises include finding remainders of dividing polynomials by (x - a), writing a remainder without dividing, finding a coefficient given a remainder, and identifying roots, integer roots, and writing polynomials with a specified root.

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