2.3 Real Zeros of Polynomial Functions

In the graph of , you can see the x intercept is -3

Therefore, f(-3)=0 so x=-3 is a zero of the function. Now we know that (x+3) is factor of the function. To find the other factor of the function, you must divide the function by (x+3). There are two ways of doing this.

1. Long Division

The process is similar to long dividing integers.

Example:

Set up the equation as you would in a regular long division problem

Divide the leading term of the dividend by the leading term of the divisor

Multiply the x by both terms in the divisor and place the product directly underneath the terms with the corresponding power

Just like regular long division, subtract the product from the terms above, leaving you with a single term

Bring down the next term, 10, and repeat the process of dividing the leading term by x, giving you -10

Multiply x and 1 by to give you the remaining product

Once again, subtract the product from above to finish the equation. Above the long division sign if your quotient,  x-10.

If the last step of subtraction does not equal zero, the equation has a remainder.

In this case, you take the number left over and put it over the divisor of the equation, giving you the remainder. 

Example:

with a remainder of -7

So the final answer is 

2. Synthetic Division

Example:

Put the coefficients at the top of a flipped division sign

Put 1, the known zero of the function, outside the box

Bring down the leading coefficient. Multiply the number by the zero and put the product below the next coefficient

Add the numbers in the column and multiply the sum by the zero. Place that product under the next coefficient

Again, add the numbers in the new column. If the last column equals 0, the solution has no remainder. In this case, there is a remainder of 12

Final Answer: