Dividing Polynomials; Remainder and Factor Theorems Section 2.4
Long Division of Polynomials Very similar to numerical division
Long Division Steps (on page 283 and handout) 1. Arrange the terms of both the dividend and the divisor in descending power. Include zeros for missing terms. 2. Divide the first term of the dividend by the first term of the divisor. The result is the first term of the quotient. 3. Multiply every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend with like terms lined up. 4. Subtract the product from the dividend. 5. Bring down the next term in the original dividend. 6. Repeat steps 2-5 until the remaining dividend (remainder) can no longer be divided by the divisor. 7. Write the remainder over the divisor as the last part of the quotient.
x 3 +3x 2 -4x-12 divided by x-2 Example 4 – Long and Synthetic Division
Can only be used to divide a polynomial by a linear binomial of the form x-r. Do not write the x’s, only the coefficients. Write r in the upper left, and the coefficients of the polynomial on the top. r has the opposite sign of the number next to x Polynomial must be in standard form. Write a zero for any missing term. Synthetic Division
Synthetic Division Steps (page 286) 1. Arrange the terms of the dividend in descending power, with a zero for any missing terms. 2. Write “r” (switch the sign) of the divisor, x-r. Put “r” in a half box. Write the coefficients of the divisor to the right (don’t forget the zeros). 3. Skip a space, draw a horizontal line, and write the leading coefficient of the dividend below the line, below where it is above. 4. Multiply “r” by the number you just wrote below the line. Write the product above the line in the next column.
Synthetic Division Steps, cont. (page 286) 5. Add the values in this new column, and write the sum below the line. 6. Repeat steps 4 and 5 until all columns are filled in. 7. The numbers below the line are the coefficients of the quotient, plus the remainder. The degree of the first term of the quotients is always one less than the degree of the first term of the dividend. 8. Write the quotient with the variable. The degree of the quotient is always one less than the degree of the dividend.
Solving Polynomials If you divide f(x) by (x-r) and receive a remainder of zero, then r is a zero of f(x), and a solution of f(x)=0. If you know one zero of f(x), you can use synthetic division to divide by (x-r) and get a lower degree polynomial. Solve this new polynomial. Any roots of this polynomial are also roots of f(x). Remember: root, zero, solution, and x-intercept all mean approximately the same thing.
Example 6 Solve the equation 3x 3 +7x 2 -22x-8=0 given that -1/3 is a root.
You-Try #6 Solve the equation 2x 3 -3x 2 -11x+6=0 given that -2 is a root.
Assignment Page 290 #1-31 every other odd, 37-41 odd