1. A few more things about graphing and zeros.

2. ± 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑎𝑑 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

3. List all of the possible rational zeros of each function.
2. 𝑓 𝑥 =2 𝑥 3 −5 𝑥 2 −10𝑥+6
Seems like busywork, huh? But if you see an x-intercept and it isn’t an integer, you can make a guess as to what rational zero (fraction) it might be.

4. Factor each and find all zeros.
2. 𝑓 𝑥 =2 𝑥 3 −5 𝑥 2 −10𝑥+6

5. Just like the imaginary roots, radical roots come in pairs of conjugates IF the coefficients of the polynomial are integers.

6. Steps to find zeros or solutions of polynomial equations (or find factors!)
List the possible rational zeros
Graph and try to identify the x-intercepts.
Do synthetic division (substitution) to see if a chosen x-intercept (r) gives a remainder of zero. If yes, continue on. If no, try a different x-intercept.
Continue with the results from step 3 to see if another x-intercept works.
Repeat with the results until you are finally to a quadratic factor.
When you get quadratic factors you can ALWAYS find the remaining zeros.

7. State the possible rational zeros for each function
State the possible rational zeros for each function. Then factor each and find all rational zeros. One zero has been given.
𝑓(𝑥)= 6𝑥 𝑥 𝑥 𝑥 𝑥 3 −6 𝑥 2 −45𝑥−13;
−3+2𝑖