1. The factors of the constant. The factors of the leading coefficient. The factors of the 12. The factors of the 2. 1, 2, 3, 4, 6, 12 1, 2 = + {1, 2, 3, 4, 6, 12,, } = 1212 3232

2. Example

3. Synthetic Division is used to quickly check, if possible rational zeros are factors. Test x = -1. Synthetic division can only be done if the degree of the divisor is 1. 16116 x + 1 = 0 x = – 1 – 1 1 – 6 6 – 5 50 Move the first number down to the bottom row. Multiply the potential zero to the bottom row number and move the product up to the next column to combine. Repeat the process until you have the remainder at the bottom of the last column.

4. + – + – + – + Alternating signs + + + + + + + Positive signs Zero can be used for any needed sign! ONLY TEST NEGATIVE RATIONAL ZEROS! ONLY TEST POSITIVE RATIONAL ZEROS!

5. 2 7 -7 -12 Lower bound -223 -4 -13 -6 14 26 -2 is not a lower bound the signs do not alternate. 2 7 -7 -12 -42 -8 -3 4 0 12 -4 is not a lower bound, but it is a zero. 2 7 -7 -12 Upper bound 3213 6 32 39 84 96 3 is an upper bound the signs are all positive.

6. Find the zeros of. zeros are x-intercepts, soooooo lets graph it! Using the possible rational zero theorem, the smallest number is -6 and the largest is 6. Set your window or Zoom 6. Count tick marks to make logical guesses! Hit Trace and type in your guess. Zeros are x = -6, x = 1, x = - ½

7. Find the zeros of. Zeros are x = 3, x = -1 WAIT!! Take another LOOK! It doesn’t cross the x-axis! What does this mean? -1 has even multiplicity. Probably 2. Why?, -1 The graph is touching or crossing the x-axis 4 times and with a 5 th degree we need at most 5 zeros. Now we will use the existing 3 rational zeros to find the other 2 zeros that are not rational numbers.

8. Find the zeros of. Zeros are x = 3, x = -1, -1 1 3 -10 -22 -7 3 12 -12 -2 -10 12 Start by factoring out the two -1 zeros. 10-3 30 Remainder 11 -13 3 13-3 0 Remainder Now the 3. 314 3 12 0 -3 Remainder Complete the square Means ZERO

9. Find the zeros of Zeros are x = 1, 1 WAIT!! Take another LOOK! It crosses the x-axis like what type of curve? A Cubic curve. 1 has odd multiplicity. Probably 3. Why?, 1 The graph is crossing the x-axis in two other locations and with a 5 th degree we need at most 5 zeros. Now we will use the existing 3 rational zeros to find the other 2 zeros that are not rational numbers.

10. Find the zeros of. Zeros are x = 1, 1 1 -3 -5 23 -24 8 11-2 1 -7 -2 16 -7 Start by factoring out all the 1’s. 16-8 0 Remainder 11 1 -8 8 -8 8 0 Remainder 110 1 -8 0 0 Remainder Solve for x. Means ZERO

11. c = f(c) 1 0 -12 6 -21 -8 4 22 16 = f(-2) When an x-term is missing, put in a zero to hold the place of the term. 1 -2 4 -25 0 -4 31 1 3 7 3 -4 21-12-36 -12-40= f(3) 4