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