1. 6.4 Solving Polynomial Equations

2. One of the topics in this section is finding the cube or cube root of a number.
A cubed number is the solution when a number is multiplied by itself three times.
A cube root “undos” the cubing operation just like a square root would.

3. Calculator Function – How to take the cube root of a number
To take the cube root of a number, press MATH, then select option 4.
Example: What is ? 24

4. Solving Polynomials by Graphing
We start getting into more interesting equations now . . .
Ex: x3 + 3x2 = x + 3
Problem: Solve the equation above, using a graphing calculator

5. Solving Polynomials by Graphing
What about something like this???
Ex: x3 + 3x2 + x = 10
Use the same principal; plug the first part of the equation in for Y1; the solution (10) for Y2; then find the intersection of the two graphs.

6. FACTORING AND ROOTS CUBIC FACTORING
Difference of Cubes
a³ - b³ = (a - b)(a² + ab + b²)
Sum of Cubes
a³ + b³ = (a + b)(a² - ab + b²)
Question: if we are solving for x, how many possible answers can we expect?
3 because it is a cubic!

7. CUBIC FACTORING EX- factor and solve
a³ - b³ = (a - b)(a² + ab + b²)
8x³ - 27 = (2x - 3)((2x)² + (2x)3 + 3²)
(2x - 3)(4x² + 6x + 9)=0
X= 3/2
Quadratic Formula

8. CUBIC FACTORING EX- factor and solve
a³ + b³ = (a + b)(a² - ab + b²)
x³ = (x + 7)(x² - 7x + 7²)
(x + 7)(x² - 7x + 49)=0
X= -7
Quadratic Formula

9. Let’s try one
Factor
a) x3-8 b) x3-125

10. Let’s try one
Factor
a) x3-8 b) x3-125

11. Let’s Try One
81x3-192=0
Hint: IS there a GCF???

12. Let’s Try One
81x3-192=0

13. Factor by Using a Quadratic Form
Ex: x4-2x2-8
= (x2)2 – 2(x2) – 8
Substitute a in for x2
= a2 – 2a – 8
This is something that we can factor
(a-4)(a+2)
Now, substitute x2 back in for a
(x2-4)(x2+2)
(x2-4) can factor, so we rewrite it as (x-2)(x+2)
Since this equation has the form of a quadratic expression, we can factor it like one. We will make temporary substitutions for the variables
So, x4-2x2-8 will factor to (x-2)(x+2)(x2+2)

14. Let’s Try One
Factor x4+7x2+6

15. Let’s Try One
Factor x4+7x2+6

16. Let’s Try One Where we SOLVE a Higher Degree Polynomial
x4-x2 = 12

17. Let’s Try One Where we SOLVE a Higher Degree Polynomial
x4-x2 = 12