T he p olynomial p roject

1. The polynomial project Done by : Fatima Abdulla Alweshahi Sara saeedAlkaabi . 11:51

2. Polynomial This Way 

3. Task1 • Task 2 • Task 3 • Task 4 • information about Polynomial

4. Task 1 : Find the polynomial that gives the following values • a. Write the system of equations in A, B, C, and D that you can use to find the desired polynomial. • 10=A • -6=A+B(x1-x0) • -17=A+B(x2-x0)+(x2-x0) (x2-x1) • 82=A+B(x3-x0)+(x3-x0)(x3-x1)+D (x3-x0) (x3-x1)(x3-x2)

5. Solve the system obtained from part a. 10 = A -6 = 10 + B (1-(-1)  2B = -16  B = -8 -17= 10 + -8 ( 2-(-1)) + ((2 – ( - 1) ( 2 – 1)  3(= - 3  (= - 1 82 =10+ -8 (5 – ( - 1 )) + - 1 (5-(-1)) (5-1) +D (5-(-1)) (5-1) (5-2) 72D= 144  D=2 A=10 B= -8 (= - 1 D =2 c. Find the polynomial that represents the four ordered pairs. P( x ) = 10 + -8 (x – ( - 1 ) + - 1 (x – ( - 1 )) ( x – 1 ) +2) x – ( - 1 )) ( x – 1 ) ( x – 2 ) =10 – 8 x - 8 – x 2 + 1 + 2 (x3 – 2x2 – x + 2 ) = 3 – x 2 – 8 x + 2x3 – 4x2 – 2 x + 4 = 2 x 3 – 5 x 2 – 10 x + f. D . Write the general form of the polynomial of degree 4 for 5 pairs of numbers. P (x) = A+B ( x – x 0) + ( x – x 0) ( x – x 1) + D ( x – x 0) ( x – x 1 ) ( x – x 2 ) + E ( x – x 0 ) ( x – x 1 ) ( x – x 2) (x-x3) = E x 4 + ( 2 – f E ) x 3 + (9 E – 5 ) x 2 + (FE – 10 ) x + (F – 10 E )

6. Task 2: Find the zeros of the polynomial found in task 1. A . Show that the 3 zeros of the polynomial found in task 1 are: First zero lies between -2 and -1 Second zero lies between 0 and 1 Third zero lies between 3 and 4. F ( - 2 ) = 1 , F ( - 1 ) = 10  mid point = - 1 . 5 F (0) = F , F ( 1 ) = - 6  = 0 . 5 F ( 3 ) = - 14 , F (4) = 15  = 3 . 5

7. B . Find to the nearest tenth the third zero using the Bisection Method for Approximating Real Zeros. F (3) = - 14 , f (4) = 15  3 . 5 F ( 3 . 5 ) = - 3 . 5 mid point F (3 . 5 ) = - 3 . 5 ، f ( 4) = 15  3 . 75 F ( 3 . 75 ) = 4 . 658 mid point F ( 3 . 5 ) = 3. 5 ، f ( 3 .75 ) = 4 . 658  3 . 625 F ( 3 . 75 ) = 4 .658 mid point F ( 3 . 5 ) = 3 . 5 ،f ( 3 . 75 ) = 4 . 658  3 . 625 F (3 . 625) = 0 . 319 F (3. 625 ) = 0 . 319 ، f ( 3 . 75 ) = 4 . 658  3 . 68f5 F ( 3 . 68 f 5 ) = 2 . 42 F ( 3 . 68f 5 ) = 2 .42 ،f ( 3 . 75 ) = 4 658  3 . 718

8. Task 3: Real World Construction You are planning a rectangular garden. Its length is twice its width. You want a walkway w feet wide around the garden. Let x be the width of the garden. A . Choose any value for the width of the walkway w that is less than 6 ft. W = 1 f t B . Write an expression for the area of the garden and walk. A = L x w = 2 w x w A=2x2ft2 Parameter = 2 (( + w ) = 2 ( 2 x + x ) = 4 x + 2 x parameter = 6 x

9. C . Write an expression for the area of the walkway only. Parameter = 2 (( + w ) = 2 (2 w + w ) = 2 ( 3 w ) = 2 ( 3 x ) = 2 6 x f t D . You have enough gravel to cover 1000ft2 and want to use it all on the walk. How big should you make the garden? 1000 = 2 w 2 1000 = 2 x 2 500 = x 2 22 . 36f t =x L = x 2 = 2 x 22 . 36 L = 44 . 72 f t

10. Task 4: Using Technology: a. Use a graphing program to graph the polynomial found in task 1 B . Make a PowerPoint to present your project and upload it on a wiki. http://mathproject121.wikispaces.com/

11. Polynomials A polynomial is made up of terms that are only added, subtracted or multiplied. A polynomial looks like this:

12. Polynomialcomes form poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms" A polynomial can have: constants (like 3, -20, or ½) variables (like x and y) exponents (like the 2 in y2), but only 0, 1, 2, 3, ... etc and they can be combined using: + addition, - subtraction, and × multiplication   ... but not division! Those rules keeps polynomials simple, so they are easy to work with!

13. Polynomial or Not? These are polynomials: 3x x - 2 -6y2 - (7/9)x 3xyz + 3xy2z - 0.1xz - 200y + 0.5 512v5+ 99w5 1 (Yes, even "1" is a polynomial, it has one term which just happens to be a constant).

14. And these are not polynomials 2/(x+2) is not, because dividing is not allowed 1/x is not 3xy-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...) √x is not, because the exponent is "½" (see fractional exponents) But these are allowed: x/2is allowed, because it is also (½)x (the coefficient is ½, or 0.5) also 3x/8 for the same reason (the coefficient is 3/8, or 0.375) √2 is allowed, because it is a constant (= 1.4142...etc)

15. There are special names for polynomials with 1, 2 or 3 terms:

16. Monomial

17. 2 • Binomial

18. 3 • Trinomial

19. ex

20. remember the names? How do you

21. How do you remember the names? Think cycles!

22. Can Have Lots and Lots of Terms Polynomials can have as many terms as needed, but not an infinite number of terms.

24. What is Special About Polynomials? Because of the strict definition, polynomials are easy to work with. For example we know that: If you add polynomials you get a polynomial If you multiply polynomials you get a polynomial So you can do lots of additions and multiplications, and still have a polynomial as the result. Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.

26. Degree The degree of a polynomial with only one variable is the largest exponent of that variable.

27. Standard Form The Standard Form for writing a polynomial is to put the terms with the highest degree first. You don't have to use Standard Form, but it helps.