Polynomial Function and Synthetic Division
- 2. A polynomial is a function of the form: Note: • an represents any real number • n represents any whole number • The degree of a polynomial with one variable is the largest exponent of all the terms. • We arrange terms of polynomials in descending order based on their degree and classify them as follows: f(x) = anxn + an−1xn−1 + . . . + a2x2 + a1x + a0
- 3. Highest Degree of 1 Example: x + 6
- 4. Highest Degree of 2 Example: x 2 + 6
- 5. Highest Degree of 3 Example: x3 + 6
- 6. Highest Degree of 4 Example: x 4 + 6
- 7. Highest Degree of 5 Example: x 5 + 6
- 8. Step5 Identify the Given Identify the polynomial function; Consider the x-values to be ( -2, -1, 0, 1, 2 )
- 9. Step5 Set up the table of values Given the polynomial function and the x-values, substitute each x-value to the function to get the corresponding y-value
- 10. Step5 Graph the coordinates When you get all the y-values, graph all of the coordinates ; The number of turning points of the graph can be determined by subtracting 1 from the leading term of the original polynomial function, if the graph did not obtain the expected number of turning points, extend the x-values until you get it
- 11. Example: f(x) = 2x4 + 3x3 - 5x2 x -2 -1 0 1 2 y -12 -6 0 0 36
- 12. Example: f(x) = 3x3 - 4x2 + 7 x -2 -1 0 1 2 y -33 0 7 6 15
- 13. Example: f(x) = 2x4 + 4x3 - 3x2 x -2 -1 0 1 2 y -12 -5 0 3 52
- 14. Example: f(x) = x3 - 2x x -2 -1 0 1 2 y -4 1 0 -1 4
- 15. Example: f(x) = 7x3 + 12x + 3 x -2 -1 0 1 2 y -77 -16 3 22 83
- 16. D i v i s i o n S y n t h e t i cS y n t h e t i c D i v i s i o n
- 17. -Is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor - Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials.
- 18. Step5 Arrange Arrange the coefficients of P(x) in descending powers of x, placing 0s for the missing terms. The leading coefficient of P(x) becomes the first entry of the third row.
- 19. Step5 Value of r Place the value of r in the upper left corner
- 20. Step5 Set it up The first number in the second row is the product of r and the number in the third row of the preceding column. The second number in the third row is the sum of the two numbers above it.
- 21. Step5 Repeat Repeat the procedure described in Step 3 until the last number in the third row is obtained.
- 22. Step5 Final Answer Write the quotient Q(x). Note that the degree of Q(x) is one less than the degree of P(x). The entries in the third row give the coefficients of Q(x) and the remainder R.
- 27. Example: (3x3 + 4x2 + 8) ÷ (x+2)
- 28. 10 – GALILEI 2015 - 2016 Julian, Ma. Mikaela Nacional, Abigail Oller, Joharah Bianca Quesada, Aleczandra Quirino, Christiane Jamila Caloocan National Science and Technology High School