Consider the given function.
f(x)=x^4-3x^3-3x^2-75x-700
We see the degree of the polynomial is 4. According to the Fundamental Theorem of Algebra, the function has four zeros. This is including any repeated zeros.
Step 2
Let's now determine the type of zeros. To do so, we will examine the number of sign changes for f(x).
There is only one sign change for the coefficients of f(x). According to Descartes' Rule of Signs, the function has one positive real zero. Next, let's write and simplify f(- x).
f(- x) = (- x)^4 - 3(- x)^3 - 3(- x)^2 - 75(- x) - 700
⇕
f(- x)=x^4+3x^3-3x^2+75x-700
As we did for f(x), we will examine the number of sign changes for f(- x).
There are three sign changes for the coefficients of f(- x). Once again, according to Descartes' Rule of Signs, the function has one or three negative real zeros. Therefore, there are two options for the types of zeros of f(x). Remember, we know that there are four zeros in total.
Option 1
Option 2
Real Zeros
1 positive
Real Zeros
1 positive
1 negative
3 negative
Imaginary Zeros
2
Imaginary Zeros
Number of Zeros
1+ 1+2=4 total
Number of Zeros
1+ 3+ =4 total
Step 3
Let's now determine the real zeros by making a table of values.
x
x^4-3x^3-3x^2-75x-700
f(x)=x^4-3x^3-3x^2-75x-700
- 4
( - 4)^4-3( - 4)^3-3( - 4)^2-75( - 4)-700
0
- 2
( - 2)^4-3( - 2)^3-3( - 2)^2-75( - 2)-700
-522
0
0^4-3( 0)^3-3( 0)^2-75( 0)-700
- 700
2
2^4-3( 2)^3-3( 2)^2-75( 2)-700
- 870
4
4^4-3( 4)^3-3( 4)^2-75( 4)-700
- 984
6
6^4-3( 6)^3-3( 6)^2-75( 6)-700
- 610
7
7^4-3( 7)^3-3( 7)^2-75( 7)-700
0
8
8^4-3( 8)^3-3( 8)^2-75( 8)-700
1068
We see above that a zero occurs at x=- 4. Let's use synthetic division to depress the polynomial.
The above are the coefficients of the depressed polynomial. Moreover, since - 4 is a root of f(x)=0, (x+4) is a factor of f(x).
f(x)=x^4-3x^3-3x^2-75x-700
⇕
f(x)=(x+4) ( x^3-7x^2+25x-175 )
From the table, we know that another zero occurs at x=7. Thus, (x-7) is a factor of f(x). We can use synthetic division again to rewrite the depressed polynomial. If you want to see how to use synthetic division with the depressed polynomial, please see the end of this exercise.
f(x)=(x+4) ( x^3-7x^2+25x-175 )
⇕
f(x)=(x+4) (x-7) ( x^2+25 )
Now the depressed is a quadratic polynomial. We can find its zeros by taking the square roots. Remember to consider the positive and negative solutions.
Since x=5i and x=-5i are zeros of the depressed polynomial, we can rewrite them as (x-5i)(x+5i).
f(x)=(x+4) (x-7) ( x^2+25 )
⇕
f(x)=(x+4) (x-7)(x-5i)(x+5i)
The four zeros are - 4, 7, 5i, and -5i.
Showing Our Work
Synthetic division
Let's use synthetic division to divide x^3-7x^2+25x-175 by (x-7).
