6.4 solving polynomial equations. one of the topics in this section is finding the cube or cube root...
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6.4 Solving Polynomial Equations
• 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.
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 ?3 13824
24
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
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.
FACTORING AND ROOTSCUBIC FACTORING
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
Difference of Cubes
Sum of Cubes
Question: if we are solving for x, how many possible answers can we expect?
3 because it is a cubic!
CUBIC FACTORINGEX- factor and solve
8x³ - 27 = 0
8x³ - 27 = (2x - 3)((2x)² + (2x)3 + 3²)
(2x - 3)(4x² + 6x + 9)=0
Quadratic Formula
3i33x X= 3/2
a³ - b³ = (a - b)(a² + ab + b²)b
axx
327
283
3 3
CUBIC FACTORINGEX- factor and solve
x³ + 343 = 0
a³ + b³ = (a + b)(a² - ab + b²)
x³ + 343 = (x + 7)(x² - 7x + 7²)
(x + 7)(x² - 7x + 49)=0
Quadratic Formula
2
377 ix
X= -7
b
axxx
7343
113
3 3
Let’s try one
• Factor
a) x3-8 b) x3-125
Let’s try one
• Factor
a) x3-8 b) x3-125
Let’s Try One
• 81x3-192=0
Hint: IS there a GCF???
Let’s Try One
• 81x3-192=0
Factor by Using a Quadratic Form
Ex: x4-2x2-8
Since this equation has the form of a
quadratic expression, we
can factor it like one. We
will make temporary
substitutions for the
variables
= (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)
So, x4-2x2-8 will factor to
(x-2)(x+2)(x2+2)
Let’s Try One
• Factor x4+7x2+6
Let’s Try One
• Factor x4+7x2+6
Let’s Try OneWhere we SOLVE a Higher Degree
Polynomial
• x4-x2 = 12
Let’s Try OneWhere we SOLVE a Higher Degree
Polynomial
• x4-x2 = 12