-2x^4+(6-2b)x^3+(6+6b)x^2+(6+8b)x+8 find for the zeros

Question

-2x^4+(6-2b)x^3+(6+6b)x^2+(6+8b)x+8

Answer 1

The zeros are the x values that make the polynomial equal to zero.  Your zeros will be in terms of b.  Since the degree of the polynomial is 4, we should have at most 4 distinct zeros.

 

What we can do first is get the coefficient of x⁴ to be 1 by dividing both sides of the equation by -2. 

 

x⁴ - (3 - b)x³ - (3 + 3b)x² - (3 + 4b)x - 4 = 0

 

 

Now we can use synthetic division to find the zeros.  The first time use synthetic division, we will find one root, as well as a factor of the polynomial.  Then we repeat this process until we end up with a quadratic factor.  When we get to that quadratic factor, we can easily use the quadratic formula on the factor to find the remaining zeros.

 

How synthetic division works is that we divide the coefficients of the polynomial by a possible zero of the polynomial.  Of course, the remainder must be zero.

 

The possible roots of the polynomial are 

 

c = ±1, ±2, and ±4.

 

 

I will give you a sample of how the division is set up including the details of the division process, in case you are new to this method.  Then I want you to apply method this to completely answer the question.

 

 

 

c |  1           (-3 + b)            (-3 - 3b)            (-3 - 4b)            -4

 

 

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      1

 

 

Step 1:  Bring the first coefficient of the polynomial under the line as you see above.  The numbers we put down under the line serve as coefficients of the quotient, with the last number serving as the remainder. 

 

Step 2:  Multiply the possible root, c, by that number under the line.

 

Step 3:  Bring the product to the next column under the second coefficient.  You will add these values and bring the sum under the line in the second column.

 

Step 4:  Multiply that sum by the possible root and place the product in the succeeding column.

 

Step 5:  Repeat steps 3-4 until you get to the last column.

 

 

You will need to try different possible roots until you get a remainder of zero.

 

 

If you get lost, let me know.