Chapter 8.8, Problem 37PPE

To determine

To describe: The way of finding the factor of the given expression $6x^5+4x^4+12x^3+8x^2+9x+6$ .

Explanation of Solution

Given info:

The given polynomial form is $6x^5+4x^4+12x^3+8x^2+9x+6$ .

Description:

Step Procedure:

1. Select the polynomial form.
2. Apply synthetic division to find the factors.
3. Choose the last term in the polynomial form, that is 6.
4. The divisible value in the given polynomial lies between $1t\text{o}6$ .
5. Do trial and error method to find the correct divisible value between $\pm 1t\text{o}6$ .
6. Choose the divisor from $\pm 1t\text{o}6$ . Take -1 as a divisor
7. Format of synthetic division is
8. Write the coefficients of x variables separately.
9. Write down the coefficients alone separately in synthetic division format.

  

10. Take the first coefficient ‘6’as such below.
11. Multiply 6 and -1 that is $-1\times 6=-6$
12. Put -6 at the bottom of 4 above the line.
13. Subtract $4-6=-2$ . The answer -2 will be at the place below -6.

  

14. Follow the same procedure for all coefficients.
15. The final resultant coefficient are obtained as

  

16. The last term ‘-9’ is consider as remainder and all other resultant coefficients are taken as quotient.
17. Do the same procedure until you get ‘0’ as a remainder with changing the divisor as
+1, -2,+2,-3,+3,-4,+4,-5,+5,-6,+6.

18. Take -2/3 as a divisor to last remainder as zero.
1. 
2. 
3. 
4. 
19. The quotients or resultant coefficients are 6, 0, 12, 0, 9, the remainder as ‘0’and foremost main thing is divisor $-\frac{2}{3}$
20. The divisor is written in the terms of factor is $x-divisor$
21. The first factor is
$\begin{array}{l}=x-divisor\\ =\left(x-\left(-\frac{2}{3}\right)\right)\\ =\left(x+\frac{2}{3}\right)\end{array}$
22. Again the polynomial can be written with help of resultant coefficients as follows:

  $\begin{array}{l}factor=6x^4+0x^3+12x^2+0x+9\\ factor=6x^4+12x^2+9\end{array}$

23. Finally, the factors of $6x^5+4x^4+12x^3+8x^2+9x+6$ is $\left(x+\frac{2}{3}\right)\left(6x^4+12x^2+9\right)$

Conclusion:

Thus, the factors of $6x^5+4x^4+12x^3+8x^2+9x+6$ is $\left(x+\frac{2}{3}\right)\left(6x^4+12x^2+9\right)$ .