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Unit 3 Study Guide Name ____________________________________________________ Objectives: Name polynomials according to their degree and number of terms. Add and subtract polynomials. Concepts: A polynomial is in standard form when terms are written so that the degrees are in descending order. Name by Number of Terms 1 Name by Degree 0 2 1 3 2 4 or more polynomial with 4 terms (5 terms, etc.) 3 4 5 6 or more 6th degree polynomial (7th degree, etc.) Examples: Write in standard form and classify each polynomial by degree and number of terms. 1. 7m2 – m + 8m4 + 6m3 2. -8x3 + 2x7 + 9x5 3. -3n3 4. 10 5. 10 – 5n2 6. 5x Objective: Add and subtract polynomials. Concepts: To add/subtract polynomials, just combine like terms. Terms are alike if they have the same variables with the same exponents. Examples: Simplify. 7. (5x2 - x - 7) + (2x2 + 3x + 4) 9. (5a2x + 3ax2 - 5x) + (2a2x - 5ax2 + 7x) 8. (5a + 9b) - (4b + 2a) 10. (x3 – 3x2y + 4xy2 + y3) – (7x3 – 9x2y + xy2 + y3) Objective: Find the zeros of function and describe end behavior without graphing. Concepts: The zeros of a function are the values at which the graph of the function touches the ____________________. Zeros are also called ______________________, ________________________, and __________________. To find the zeros of a factored function, set each factor equal to __________ and solve. You can find the zeros of non-factored functions by using your calculator (2nd, trace, 2:zero, …) The end behavior of a function can be determined by its degree and the sign of the leading coefficient. Even (degree) functions have end behavior like _________________________. Odd (degree) functions have end behavior like __________________________. Examples: Find the zeros and describe the end behavior without graphing! 11. f(x) = (x – 2)2(x+3)(3x – 1) 12. f(x) = x(x + 5)(x – 2) zeros: _____________ zeros: _____________ degree: ____________ degree: ____________ end behavior: __________________ end behavior: __________________ 13. f(x) = -2x2(x +1)(2x + 3)3 14. f(x) = -3(x – 5)3(2x – 1)4 zeros: _____________ zeros: _____________ degree: ____________ degree: ____________ end behavior: __________________ end behavior: __________________ Objective: Multiply polynomials and divide polynomials using long division. Examples: Multiply. 15. (x + 5)(x – 2) 16. (4x – 5)(2x – 3) 17. (x + 3y)(3x + 4y) 18. (x – 7)(6x2 – x + 5) Examples: Use long division to divide. 19. (2x3 + 4x2 – 5) ÷ (x + 3) 20. (12x3 – 11x2 + 9x + 18) ÷ (4x + 3) Objective: Solve polynomial equations by factoring. Concepts: Complex number: a number than can be written in the form a + bi (includes all real and imaginary numbers) √−1 = 𝑖 Sum/difference of cubes: a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) Examples: Find all complex solutions by factoring. 21. x3 – 2x2 – 9x + 18 = 0 22. 81x4 – 16 = 0 23. 4x6 – 20x4 + 24x2 = 0 24. 2x5 + 24x = 14x3 25. x3 – 3x2 = 0 26. 2x4 – 26x2 + 72 = 0 Objective: Solve quadratic equations using quadratic formula. Write imaginary solutions in simplest radical form. Concepts: Quadratic formula: Examples: Use quadratic formula to solve. 27. 2x2 – 6x + 5 = 0 28. 8x2 – 4x + 5 = 0 29. 2x2 – 6x + 7 = 0 30. -7x2 + 12x = 10 Objective: Find all complex solutions of a polynomial. Concepts: First, try factoring. If it doesn’t factor (or you can’t figure out how to factor it) graph it to find a real solution. Then use long division to find the remaining factor. Set each factor equal to zero and solve (by quadratic formula or taking the square root). Examples: Find all complex solutions. 31. x4 – 14x + 45 = 0 32. x3 + 3x2 – 14x – 20 = 0 33. x3 – 2x2 + 3x – 6 = 0 34. x3 – 1 = 0 Objective: Convert quadratic functions to vertex form by completing the square. Examples: Convert to vertex form and state the vertex. 35. y = x2 + 16x + 71 36. y = x2 − 2x – 5 37. y = 2x2 + 36x + 170 38. y = -6x2 – 12x – 13