1. Warmup
Solve: 𝑥 4 −27𝑥=0

2. 5-7&5-8 Roots and ZEROs
Determine the number of solutions for a polynomial equation.
Find the solutions of a polynomial equation.

4. Recall that all real numbers can be written as complex
𝑥 3 +2 𝑥 2 +6=0
4𝑥 4 −3 𝑥 3 +5𝑥−6=0
−2𝑥 5 −3 𝑥 2 +8=0

5. In the last section we learned how to figure out if a binomial (x – r) was a factor of a polynomial. Now we need to learn how to recognize which roots to try.
Your graphing calculator can help with that.

6. Repeated Roots: You can recognize an even number of repeated roots from a graph of the polynomial if the curve is only tangent to the x-axis at a value x = r, and doesn’t cross at x = r (bounces off the x-axis). You will need to use them an even number of times to get factors or zeros.
p. 363 Find all the zeros of each function.

7. p. 363 Find all the zeros of the function.

9. p Write a polynomial function of least degree with integral coefficients that have the given zeros.

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

11. p. 370 List all of the possible rational zeros of each function.
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.

12. 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.
If asked to write as factored form, write as (x – r) for all the roots.

13. p.370 Find all of the zeros of the function.

14. p.370 Find all of the zeros of the function.

15. Just like the imaginary roots, radical roots come in pairs of conjugates
IF the coefficients of the polynomial are integers.
p372 Write a polynomial of least degree with integral coefficients that has the given zeros.