SOLVING POLYNOMIALS

By Nathan, Cole, Austin and Luke

AUSTIN-SOLVING CUBIC EQUATIONS

A cubic equation is an expression of the form ax^3+bx^2+cx+d.

Here are some examples 8x^3+125 2x^3+2x^2-12x

Here is how you would solve one of these problems 2x^3+2x^2-12x

2x(x^2+x-6) *take out a 2x from everything[divide]*

2x(x+3)(x-2) *factor (x^2+x-6)* X=-3, X=2, X=0 *now find out what x equals

[x-2, x=2]*

AUSTIN-SOLVING CUBIC EQUATIONS

To solve 8x^3 + 125 here is what you do First you take the cubed root of the equation and put

it into the first parenthesis as so. (2x+5)

Then you put x^2 into the first part of the next parenthesis, flip the sign from the first equation and put (ax + b) next, then you ADD the square root of (b) from your first parenthesis. You should get something like this. (x^2 – 10x + 25)

Now when you get both of those you just put both parenthesis next to each other and you’re done! (2x+5)(x^2-10x+25)

AUSTIN-SOLVING CUBIC EQUATIONS

Lets do a problem! Who would like to come up?

(100x^3+13,824) 5X^3+25x^2-250x

NATHAN-DIVIDING POLYNOMIALS

Divide 3-5+4 by 3x+1

 Divide 3x3 – 2x2 + 3x – 4 by x – 3

3x^2+7x+24+68/x-3

Long Shlong Synthetic

NATHAN-DIVIDING POLYNOMIALS

Synthetic Division1. Carry down the leading coefficient2. Multiply the number on the left and

carry it to the next column3. Add down the column4. Multiply the number on the left and

carry it to the next column5. Repeat until you get a remainder/0

NATHAN-DIVIDING POLYNOMIALS

Long Division1. Set up the problem2. Divide the first two numbers by the

divisor3. Move the next number down next to

your remainder4. Divide that by the divisor5. Repeat until you end up with a

remainder or 0

COLE-FACTORING POLYNOMIALS When factoring polynomials, you are

finding numbers or polynomials that divide out evenly from the original polynomials. But in the case of polynomials, you are dividing numbers and variables out of expressions, not just dividing numbers out of numbers.

COLE-FACTORING POLYNOMIALS So lets try it!!! 3x-12

So the only thing that you can take out of this problem would be a three. 3x-12=3( )

Next you moved the three to the other side, and when you divide 3 out of 3x you get x so you put that in the parentheses. 3x-12=3(x )

And when you divide the -12 by three you get negative 4 and you place that into the parentheses and then that’s your final answer!!!1 3(x - 4)

COLE-FACTORING POLYNOMIALS So now you do some!!!!

12y^2 – 5y 7x - 7

LUKE-PASCAL’S TRIANGLE

ROWS The 1st number in each row represents

the number row that it is. (Remember that the 1 at the beginning of each row is the 0th number)

The 1 at the very top of the triangle is the 0th row.

1,3,3,1is the 3rd row. The 1st number is 3, meaning the 3rd row.

1,5,10,10,5,1 is the 5th row. The 1st number is 5, meaning that it is the 5th row.

LUKE-PASCAL’S TRIANGLE

Sums of the Rows If you do 2^4 power, that will equal the

sum of the 4th row. 1+4+6+4+1=16 (4th row) 2^4=16 (4th row) 1+6+15+20+15+6+1=64 (6th row) 2^6=64 (6th row)

LUKE-PASCAL’S TRIANGLE

The Hockey System If you take however many diaganals of

numbers, add them up, the sum of the numbers is the number that is diaganally downward.

Take 1, 6, 21, and 56 for example. Circle those numbers and at 56, turn the opposite direction, going diaganally down and the number 84 is the sum. Here is a picture to represent what I am talking about.

Unit 2Be ready to go when the bell rings!

Example 1: Name according to degree and number of terms.

a) 2x2 + 3x + 1 b) 2x4

Unit 2Example 2: Factor.

8x3 – 27

Unit 2Example 3: Solve.

x4 – 2x2 – 48 = 0

Unit 2Example 4: Divide x3 – 7x + 12 by x + 3

Unit 2Example 5:

Completely factor f(x) = x3 – -x2 - 4x + 4 given x-1 is a factor

Unit 2Example 6: Expand (x + 2y)3

Unit 2Example 7: Sketch graph. y = x4 – 3x2 + x – 6

Domain: End Behavior: max:x +, y Range: min: x -, y   _____ # complex_____# real_____ # imaginary

Unit 2Example 8: Write a function in standard

form given the roots: 3, -5i

Unit 2Example 9:

  Factors x-intercepts Solutions

Zeros

Standard Form

  9) 

   

     

   

 

x3 + 3x2 – x - 3

MSL (Common

Exam)

MSL (Common

Exam)

MSL (Common

Exam)

MSL (Common

Exam)

Want more practice???

Go to Mrs. Bell’s website.