1. NCTM Standards: 2 & 6

2. Appreciation The increase of value of an item over a period of time. The formula for compound interest can be used to find the value after appreciation. Where P is the original amount invested r is the interest rate or rate of return, t is the time invested (in years) A is the value after appreciation

3. Example 1 $1000 invested for 18 years at an unknown interest rate $500 invested for 10 years at an unknown interest rate $1000 invested for 5 years at an unknown interest rate The total value of the college fund To simplify things, let x = 1+r = + +

4. Remember: x = 1+r So the actual value to evaluate is 1.1225 = + + The total value of the college fund is $10,038.33 after 18 years. How much was the total initial investment? This expression is called a polynomial in one variable.

5. The degree of a polynomial in one variable is the greatest exponent of its variableS. = + + Zeros of a function: the values for x for which f(x) = 0. (These are also the x-intercepts when the function is graphed.) Leading coefficient: the coefficient of the variable with the greatest exponent.

6. Example 2 a.The value of the highest exponent is: The leading coefficient is: 3 (degree) 1 b. Evaluate at x = 4 to see if the value of the function is zero: Since the resulting value of the function is zero, 4 is a zero of f(x). Meaning the graph will cross or touch the x-axis at 4.

7. Polynomial Equation The term for the result of replacing f(x) with zero. A polynomial function that can be graphed A polynomial function that can be solved. Root: a solution for a polynomial equation. (this term is interchangeable with the term zero) A root can be a real number or an imaginary number such as 3i.

8. The x-intercepts represents the real solutions; imaginary solutions cannot be determined with a graph; you can determine how many imaginary solutions there are, but not what they are.

9. Example 3 Work backwards:If the roots are 2, 4i, & -4i there must have been a product of three linear factors that were equal to zero. Distribute the products & simplify. What is the special name for the last two factors? What is true about them? conjugates The imaginary part will always cancel out. This is the equation with the least degree with the given roots. b. The equation is an odd degree; it crosses the x-axis one time.

10. Example 4 The degree of the equation indicates how many complex roots (or solutions) an equation has. All 4 roots are real 2 real roots and 2 imaginary roots All 4 roots are imaginary Solve by factoring: There are 4 complex roots. The possibilities of these root are:

11. Example 4 Solve by factoring: Multiply the leading coefficient and the constant List all the factors of 36: 136 218 313 4 9 6 Look for a set of factors that will add or subtract to obtain the middle term (- 35 ) Rewrite the original problem as four parts ( )

12. These are the imaginary solutions; we can only guess where they are located.

13. a = - 16 b = 232 c = -112

14. HW: Page 209