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Polynomial Factoring Techniques and Division Methods

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Learn how to factor polynomials using the sum and difference of cubes, factor theorem, synthetic division, and long division. Practice examples included.

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Polynomial Factoring Techniques and Division Methods

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  1. Section 7.3 Products and Factors of Polynomials

  2. Factoring the Sum and Difference of Two Cubes

  3. Factor Theorem • (x-r) is a factor of the polynomial expression that defines the function P if and only if r is a solution of P(x) = 0, that is, if and only if P(r) = 0. • In other words: • Set x-r = 0 • Solve for x. x = r • Plug this r in for every x in the original polynomial • Simplify • If you get 0 then (x – r) IS a factor of the polynomial

  4. Synthetic Division

  5. Let’s look at how to do this using the example: In order to use synthetic division these two things must happen: #1 There must be a coefficient for every possible power of the variable. #2 The divisor must have a leading coefficient of 1.

  6. Step #1: Write the terms of the polynomial so the degrees are in descending order.

  7. Step #2: Write the constant a of the divisor x- a to the left and write down the coefficients.

  8. Step #3: Bring down the first coefficient, 5.

  9. Add the column Step #5: After multiplying in the diagonals, add the column.

  10. 15*3 = 45 Add Multiply the diagonals, add the columns.

  11. Add Columns Add Columns Add Columns Add Columns Step #7: Repeat the same procedure as step #6.

  12. The quotient is: Remember to place the remainder over the divisor.

  13. Step #8: Write the quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

  14. Long Division

  15. Divide a polynomial by a monomial.

  16. Divide a polynomial by a monomial.

  17. Divide the following numbers. Check

  18. Divide the following numbers (Long Division) Remainder

  19. Dividing a polynomial by a polynomial(Long Division) JUST WATCH THIS ONE! Check

  20. Dividing a polynomial by a polynomial(Long Division) JUST WATCH THIS ONE! Check

  21. Dividing a polynomial by a polynomial(Long Division) JUST WATCH THIS ONE! Check

  22. Dividing a polynomial by a polynomial(Long Division) JUST WATCH THIS ONE! Check

  23. Dividing a polynomial by a polynomial (Long Division) NOW YOU WRITE THIS ONE

  24. Dividing a polynomial by a polynomial (Long Division)

  25. Dividing a polynomial by a polynomial (Long Division) NOW YOU WRITE THIS ONE

  26. Dividing a polynomial by a polynomial (Long Division)

  27. Homework Problems 1. 2. 3. 4.

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