ment: "December 15 x December 15 & 16 - Aim: How d x | Mathway | Calculus Problem Solv X + etails?pli=1 c. x + 2x + 3.x*= 0 d. x + 32 =0 4. Visual Thinking Use a graph to explain why a cubic equation with real coefficients must have at least one real root. 5. Visual Thinking Use a graph to explain why a quartic equation with real coefficients does not necessarily have any real roots. WRITTEN EXERCISES Tell whether the statements in Exercises 1-8 are true or false. Justify your answers. (All equations mentioned have real coefficients.) A 1. Some cubic equations have no real roots. 2. Every polynomial equation has at least one real root. 3. The roots of a certain quartic equation are + 7, 0, and 1 + i. 4. The roots of a certain fifth-degree equation are - 3, 4, 1 - V2, and + i. 5. It is possible for the graph of a cubic function to be tangent to the x-axis at x = -2, x = 1, and x = 6. 6. No polynomial equation can have an odd number of imaginary roots. 7. Suppose P(x) is a polynomial with rational coefficients, and b is rational but b ;; irrational. If Vb is a root of the equation P(x) = 0, then - Vb is also a 8. if a + bi, b # 0, is a root of the polynomial equation P(x) = 0, then the equation must have an even number of roots. In Exercises 9-12, find the sum and product of the roots of the given equation. 9. 4x2- 3x +6=0 10. 613 - 9x2 + x = 0 11. 3x3 + 5x2- x - 2=0 12. x4 - 4x2 = 5 In Exercises 13-16, find a quadratic equation with integral coefficients that has the given roots. 13. 1 + i 14. 4 + V3 15. 3 + V2 16. 1= iV2 3 Polynomial Functions 89