4 ESO Academics - Unit 03 - Exercises 4.3.5 - Factorizing Polynomials. Algebraic Fractions.
- 1. UNIT 03 December 1. Factorize these polynomials a) 𝒙𝒙𝟐𝟐 − 𝟐𝟐𝟐𝟐 − 𝟑𝟑 b) 𝟓𝟓𝒙𝒙𝟑𝟑 − 𝟓𝟓𝟓𝟓 c) 𝒙𝒙𝟑𝟑 − 𝟐𝟐𝒙𝒙𝟐𝟐 − 𝟓𝟓𝟓𝟓 + 𝟔𝟔 d) 𝒙𝒙𝟑𝟑 + 𝟒𝟒𝒙𝒙𝟐𝟐 + 𝟓𝟓𝟓𝟓 + 𝟐𝟐 e) 𝒙𝒙𝟒𝟒 − 𝟑𝟑𝒙𝒙𝟐𝟐 + 𝟐𝟐𝟐𝟐 Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.5.1
- 2. UNIT 03 December 2. Write the roots of these factorizing polynomials: f) 𝒙𝒙𝟓𝟓 − 𝟓𝟓𝒙𝒙𝟑𝟑 + 𝟒𝟒𝟒𝟒 g) 𝒙𝒙𝟑𝟑 − 𝟒𝟒𝒙𝒙𝟐𝟐 + 𝟒𝟒𝒙𝒙 − 𝟏𝟏𝟏𝟏 h) 𝒙𝒙𝟒𝟒 − 𝟏𝟏𝟏𝟏𝒙𝒙𝟑𝟑 + 𝟑𝟑𝟓𝟓𝒙𝒙𝟐𝟐 − 𝟓𝟓𝟓𝟓𝟓𝟓 + 𝟐𝟐𝟐𝟐 a) (𝟐𝟐𝒙𝒙 + 𝟏𝟏) ∙ (𝒙𝒙 − 𝟐𝟐) b) (𝒙𝒙 − 𝟑𝟑) ∙ (𝒙𝒙 + 𝟓𝟓) Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.5.2
- 3. UNIT 03 December 3. Write the polynomial as product of other two polynomials in two different ways: 𝑷𝑷(𝒙𝒙) = 𝟔𝟔𝒙𝒙𝟐𝟐 + 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟏𝟏𝟏𝟏 4. Factorize these polynomials: c) 𝒙𝒙𝟐𝟐 ∙ (𝒙𝒙𝟐𝟐 + 𝟐𝟐) ∙ (𝒙𝒙 + 𝟑𝟑) d) (𝒙𝒙 − 𝟏𝟏)𝟑𝟑 ∙ (𝟑𝟑𝟑𝟑 + 𝟐𝟐) ∙ (𝒙𝒙 + 𝟒𝟒) a) 𝒙𝒙𝟑𝟑 − 𝟑𝟑𝒙𝒙𝟐𝟐 + 𝟑𝟑𝟑𝟑 − 𝟏𝟏 Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.5.3
- 4. UNIT 03 December b) 𝒙𝒙𝟑𝟑 − 𝒙𝒙𝟐𝟐 − 𝒙𝒙 + 𝟏𝟏 c) 𝒙𝒙𝟑𝟑 − 𝟕𝟕𝒙𝒙 − 𝟔𝟔 d) 𝒙𝒙𝟑𝟑 − 𝒙𝒙𝟐𝟐 − 𝟏𝟏𝟏𝟏𝒙𝒙 + 𝟐𝟐𝟒𝟒 e) 𝒙𝒙𝟒𝟒 − 𝟐𝟐𝟎𝟎𝒙𝒙𝟐𝟐 + 𝟔𝟔𝟒𝟒 f) 𝒙𝒙𝟒𝟒 + 𝟔𝟔𝒙𝒙𝟑𝟑 + 𝟏𝟏𝟏𝟏𝒙𝒙𝟐𝟐 + 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟒𝟒 Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.5.4
- 5. UNIT 03 December 5. Factorize these polynomials: g) 𝒙𝒙𝟒𝟒 + 𝟔𝟔𝒙𝒙𝟑𝟑 − 𝟓𝟓𝟓𝟓𝟓𝟓 − 𝟖𝟖𝟖𝟖 a) 𝟑𝟑𝟎𝟎𝒙𝒙𝟐𝟐 + 𝒙𝒙 − 𝟏𝟏 b) 𝟓𝟓𝒙𝒙𝟐𝟐 − 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝟑𝟑𝟑𝟑 c) 𝟐𝟐𝒙𝒙𝟑𝟑 + 𝟏𝟏𝟏𝟏𝒙𝒙𝟐𝟐 + 𝟏𝟏𝟏𝟏𝟏𝟏 d) 𝟕𝟕𝟓𝟓𝒙𝒙𝟑𝟑 − 𝟑𝟑𝟑𝟑 Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.5.5
- 6. UNIT 03 December 6. Tell if these factions are algebraic fractions or not: 𝒂𝒂) 𝟐𝟐 𝒙𝒙 + 𝟏𝟏 𝒃𝒃) 𝒙𝒙 + 𝟏𝟏 𝟐𝟐 𝒄𝒄) −𝟐𝟐𝟐𝟐 + 𝟓𝟓 𝟑𝟑 𝒅𝒅) −𝟐𝟐𝟐𝟐 + 𝟓𝟓 𝟑𝟑𝒙𝒙 7. Check if these algebraic fractions are equivalents: 𝒂𝒂) 𝟐𝟐𝒙𝒙 + 𝟏𝟏 𝒙𝒙 − 𝟑𝟑 𝒂𝒂𝒏𝒏𝒏𝒏 𝒙𝒙 − 𝟑𝟑 𝟐𝟐𝟐𝟐 + 𝟏𝟏 𝒃𝒃) 𝒙𝒙 + 𝟐𝟐 𝒙𝒙 + 𝟏𝟏 𝒂𝒂𝒏𝒏𝒏𝒏 𝒙𝒙 + 𝟓𝟓𝒙𝒙𝟐𝟐 + 𝟔𝟔 𝒙𝒙𝟐𝟐 + 𝟒𝟒𝟒𝟒 + 𝟑𝟑 e) (−𝟑𝟑𝒙𝒙𝟐𝟐 + 𝒙𝒙 + 𝟐𝟐)𝟐𝟐 f) 𝟏𝟏𝟖𝟖𝒙𝒙𝟑𝟑 + 𝟔𝟔𝒙𝒙𝟐𝟐 − 𝟓𝟓𝟓𝟓𝟓𝟓 + 𝟏𝟏𝟏𝟏 Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.5.6
- 7. UNIT 03 December 8. Which algebraic fraction we need to obtain 𝑪𝑪(𝒙𝒙) when we multiply by 𝑨𝑨(𝒙𝒙)? 𝑨𝑨(𝒙𝒙) = 𝟓𝟓𝟓𝟓 + 𝟏𝟏 𝟐𝟐𝟐𝟐 𝒂𝒂𝒏𝒏𝒏𝒏 𝑪𝑪(𝒙𝒙) = 𝟕𝟕𝟕𝟕 + 𝟑𝟑 𝒙𝒙 − 𝟏𝟏 9. Solve 𝒂𝒂) 𝟒𝟒 𝒙𝒙 + 𝟏𝟏 + 𝟐𝟐 𝒙𝒙 + 𝟐𝟐 𝒃𝒃) 𝟏𝟏 𝒙𝒙 + 𝟓𝟓 − 𝟕𝟕 𝒙𝒙 − 𝟐𝟐 𝒄𝒄) −𝟑𝟑 𝒙𝒙 − 𝟏𝟏 + 𝟖𝟖 𝒙𝒙 − 𝟑𝟑 𝒅𝒅) −𝟑𝟑 𝒙𝒙 + 𝟔𝟔 − 𝟗𝟗 𝒙𝒙 − 𝟏𝟏 Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.5.7
- 8. UNIT 03 December 10. Solve 𝒂𝒂) 𝟒𝟒 𝒙𝒙 + 𝟏𝟏 ∙ 𝒙𝒙 + 𝟐𝟐 𝟐𝟐 𝒃𝒃) −𝟑𝟑 𝒙𝒙 − 𝟏𝟏 ÷ 𝒙𝒙 − 𝟑𝟑 𝒙𝒙 𝒄𝒄) 𝒙𝒙 − 𝟓𝟓 𝟐𝟐𝒙𝒙𝟐𝟐 + 𝒙𝒙 − 𝟑𝟑 ∙ 𝒙𝒙𝟐𝟐 − 𝟏𝟏 𝟑𝟑𝒙𝒙𝟐𝟐 𝒅𝒅) 𝒙𝒙 𝟐𝟐𝒙𝒙𝟐𝟐 + 𝒙𝒙 − 𝟏𝟏 ÷ 𝒙𝒙𝟐𝟐 𝟐𝟐𝒙𝒙 − 𝟏𝟏 Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.5.8
- 9. UNIT 03 December 11. Solve 𝒂𝒂) � 𝟏𝟏 𝒙𝒙 − 𝟐𝟐 − 𝒙𝒙 − 𝟑𝟑 𝒙𝒙𝟐𝟐 − 𝟒𝟒 � ∙ 𝒙𝒙 + 𝟐𝟐 𝒙𝒙 − 𝒙𝒙 𝟐𝟐 𝒃𝒃) � 𝟔𝟔 𝟏𝟏 − 𝒙𝒙 − 𝟓𝟓𝒙𝒙 𝒙𝒙 − 𝟏𝟏 � ÷ 𝒙𝒙𝟐𝟐 − 𝟏𝟏 𝟐𝟐 + 𝟑𝟑 𝒙𝒙 𝒄𝒄) �𝒙𝒙 + 𝟏𝟏 + 𝒙𝒙𝟐𝟐 𝟏𝟏 − 𝒙𝒙 � ÷ �𝟏𝟏 − 𝒙𝒙 𝟏𝟏 + 𝒙𝒙 ∙ 𝒙𝒙 + 𝟏𝟏 𝒙𝒙𝟑𝟑 � + 𝟒𝟒 𝒙𝒙𝟐𝟐 − 𝟏𝟏 Axel Cotón Gutiérrez Mathematics 4º ESO Exercises – 4.3.5.9