Section 2.4 Dividing Polynomials; Remainder and Factor Theorems

Long Division of Polynomials and The Division Algorithm Key Vocabulary: Dividend Divisor Quotient Remainder

Long Division of Polynomials Solution is:

Long Division of Polynomials Solution is:

Long Division of Polynomial with Missing Terms Quotient Solution is: Remainder You need to leave a hole when you have missing terms. This technique will help you line up like terms. See the dividend above.

Do Now Please Divide using Long Division.

Example 5 Divide using Long Division.

Dividing Polynomials Using Synthetic Division Vocabulary: Synthetic Division

Synthetic Division To Divide a polynomial by x - c Arrange the polynomial in descending powers, with a 0 coefficient for any missing term. Write c for the divisor, x – c. To the right, write the coefficients of the dividend. Write the leading coefficients of the dividend on the bottom row. Multiply c times the value in the bottom row. Write the product in the next column in the second row. Add the values in the new column, writing the sum in the bottom row. Repeat this series of multiplications and additions until all columns are filled in. Use the numbers in the LAST row to write the quotient, plus the remainder above the divisor. The degree of the fist term of the quotient is one less than the degree of the first term of the dividend. The final value in this row is the remainder.

↓ Bring down 1 ↓ Add ↓ Add ↓ Add Multiply 3 and 1 Quotient Remainder Multiply 3 and 7 Multiply 3 and 16

Comparison of Long Division and Synthetic Division of x3 + 4x2 - 5x + 5 divided by x - 3 List at least 3 things that you notice about the relationship between Long Division and Synthetic Division.

Steps of Synthetic Division dividing 5x3 + 6x + 8 by x + 2 Quotient Remainder

Divide using synthetic division. Example 7 Divide using synthetic division. Quotient Remainder

Using Synthetic Division instead of Long Division Notice, that the divisor of all the Synthetic Division problems we have done have a degree of 1. Thus:

The Remainder Theorem If you are given the function f(x) = x3 - 4x2 + 5x + 3 and you want to find f(2), then the remainder of this function when divided by x - 2 will give you f(2). Remainder

Factor Theorem using the Remainder Theorem

They are the same!!!!!

Example 9 Use synthetic division and the remainder theorem to find the indicated function value. They are the same!!!!!

Solve the equation 2x3 - 3x2 - 11x + 6 = 0 shows that 3 is a zero of f(x) = 2x3 - 3x2 - 11x + 6. The factor theorem tells us that x - 3 is a factor of f(x). So we will use both synthetic division and long division to show this and to find another factor. Another factor

So that PROVES -2 IS a zero! AND x + 2 IS a factor of the dividend! Example 11 Solve the equation 5x2 + 9x – 2 = 0 given that -2 is a zero of f(x)= 5x2 + 9x - 2 So that PROVES -2 IS a zero! AND x + 2 IS a factor of the dividend!

AND x - 5 IS a factor of the dividend! Example 12 Solve the equation x3 - 5x2 + 9x - 45 = 0 given that 5 is a zero of f(x)= x3 - 5x2 + 9x – 45. So that PROVES 5 IS a zero! AND x - 5 IS a factor of the dividend!

Review Time!!!! Add these problems to your notes paper to help you review! Additional Practice problems can be found on page 344 – 346 problems 47-81

(a) (b) (c) (d) (d)

(a) (b) (c) (d) (b)