CUBIC POLYNOMIAL AND ITS ZEROS

                               CUBIC POLYNOMIAL 

                       
A cubic polynomial is the polynomial which is of degree three polynomial. The highest exponent of the variable is 3. For example $P(x)=3x^3+2x^2+5x+5$. It's a cubic polynomial.
The standard cubic function is $f(x)=x^3$ and its graph is symmetric about the origin as shown below: 

  This graph goes up as move right. It means function is increasing for its domain.
Graph of other cubic polynomial can be found using the transformation of the graph of standard cubic function. The points where the graph intersects the x-axis are known as zeros of the polynomial.
A cubic polynomial have three zeros and at least one zero is real. 

FACTOR THEOREM:  If a be the zero of a polynomial p(x) such that p(a)=0 then (x-a) is a factor of p(x). 
REMAINDER THEOREM: If we divide a polynomial p(x) by (x-a) then the remainder is equal to p(a) i.e. remainder is equal to the number or expression obtained by substituting x=a in the given polynomial.
Example: Find the remainder when $p(x)=2x^3+3x^2-5x+5$ is divided by $x-2$. 
Solution: To get the remainder, we will plug in 2 for x in the given polynomial, we get 
                $p(2)=2\times 2^3+3\times 2^2-5\times 2+5=23$
Hence, remainder will be 23.
Remainder theorem and factor theorem are very important in finding the factors of a polynomial of greater than 2. 

Synthetic Division Method:  However, we can divide a polynomial by (x-a) using a long division method but long division method is a little time consuming. To save time we user other technique such as Synthetic Division Method to divide a polynomial by (x-a). 
Consider the above polynomial  $p(x)=2x^3+3x^2-5x+5$ and we will divide it be $x-2$. 
Step 1: We will list the coefficients as below 
                        2     3     -5     5
Step 2: We will write 2 which is obtained from $x-2=0$ in the left of a vertical line as shown below
           

Step 3: We will write first number below the horizontal line as shown below

Step 4: Now multiply divisor 2 and the 2 which under the line and write product below 3 the second coefficient above the horizontal line as shown below

Step 5: Add 3 and 4 and write the sum under the line as shown below

Step 6: Multiply the divisor 2 and 7 and write the product 14 under the next number -5 and above the line as shown below

Step 7: Add 14 and -5 and write the sum under the horizontal line as shown below

Step 8: Similarly multiply divisor 2 and 9 and write product under the next number 5 and add column wise. Write the sum under the horizontal line. The last number i.e. left most number below the horizontal line is the remainder.
As shown below

Hence, remainder is 23.
Check out this video, it will help you.

Using the above steps we can divide any polynomial by (x-a)
Try these problems.
Use synthetic division to divide the given cubic polynomials.
1. Divide $3x^3+4x^2-6x-9$  by $x-1$ and find the remainder.
2. Divide $4x^3-5x^2+8x-9$ by $x-3$ and find the remainder. State whether (x-3) is a factor of the given polynomial. If (x-3) is a factor of the given polynomial then find other factors too.