is one of the prime factors of the polynomial
<h3>How to factor the expression?</h3>
The question implies that we determine one of the prime factors of the polynomial.
The polynomial is given as:
![x^8 - 3x^6 + x^4 - 2x^3 - 1](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201)
Expand the polynomial by adding 0's in the form of +a - a
![x^8 - 3x^6 + x^4 - 2x^3 - 1 = x^8 -2x^7 + 2x^7 - 4x^6 +x^6 + 2x^5 -2x^5- 3x^4 + 4x^4 + 2x^3 -6x^3+2x^3- x^2 -3x^2 +4x^2-2x+2x-1](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201%20%3D%20x%5E8%20-2x%5E7%20%2B%202x%5E7%20-%204x%5E6%20%2Bx%5E6%20%2B%202x%5E5%20-2x%5E5-%203x%5E4%20%2B%204x%5E4%20%2B%202x%5E3%20-6x%5E3%2B2x%5E3-%20x%5E2%20%20-3x%5E2%20%2B4x%5E2-2x%2B2x-1)
Rearrange the terms
![x^8 - 3x^6 + x^4 - 2x^3 - 1 = x^8 -2x^7 + 2x^5 - 3x^4 + 2x^3 - x^2 + 2x^7 - 4x^6 + 4x^4 -6x^3+4x^2-2x+x^6-2x^5+2x^3-3x^2+2x-1](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201%20%3D%20x%5E8%20-2x%5E7%20%2B%202x%5E5%20-%203x%5E4%20%2B%202x%5E3%20-%20x%5E2%20%2B%202x%5E7%20-%204x%5E6%20%2B%204x%5E4%20-6x%5E3%2B4x%5E2-2x%2Bx%5E6-2x%5E5%2B2x%5E3-3x%5E2%2B2x-1)
Factorize the expression
![x^8 - 3x^6 + x^4 - 2x^3 - 1 = x^2(x^6-2x^5+2x^3-3x^2+2x-1) + 2x(x^6-2x^5+2x^3-3x^2+2x-1) + 1(x^6-2x^5+2x^3-3x^2+2x-1)](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201%20%3D%20x%5E2%28x%5E6-2x%5E5%2B2x%5E3-3x%5E2%2B2x-1%29%20%2B%202x%28x%5E6-2x%5E5%2B2x%5E3-3x%5E2%2B2x-1%29%20%2B%201%28x%5E6-2x%5E5%2B2x%5E3-3x%5E2%2B2x-1%29)
Factor out x^6-2x^5+2x^3-3x^2+2x-1
![x^8 - 3x^6 + x^4 - 2x^3 - 1 = (x^2+2x + 1)(x^6-2x^5+2x^3-3x^2+2x-1)](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201%20%3D%20%28x%5E2%2B2x%20%2B%201%29%28x%5E6-2x%5E5%2B2x%5E3-3x%5E2%2B2x-1%29)
Express x^2 + 2x + 1 as a perfect square
![x^8 - 3x^6 + x^4 - 2x^3 - 1 = (x+1)^2(x^6-2x^5+2x^3-3x^2+2x-1)](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201%20%3D%20%28x%2B1%29%5E2%28x%5E6-2x%5E5%2B2x%5E3-3x%5E2%2B2x-1%29)
Expand the polynomial by adding 0's in the form of +a - a
![x^8 - 3x^6 + x^4 - 2x^3 - 1 = (x+1)^2(x^6- 2x^5+x^4-x^4-x^3 +x^3-2x^3-x^2 -2x^2 +x+x - 1)](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201%20%3D%20%28x%2B1%29%5E2%28x%5E6-%202x%5E5%2Bx%5E4-x%5E4-x%5E3%20%2Bx%5E3-2x%5E3-x%5E2%20-2x%5E2%20%2Bx%2Bx%20-%201%29)
Rearrange the terms
![x^8 - 3x^6 + x^4 - 2x^3 - 1 = (x+1)^2(x^6- 2x^5+x^4-x^3-x^4-2x^3-x^2+x+x^3-2x^2 +x - 1)](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201%20%3D%20%28x%2B1%29%5E2%28x%5E6-%202x%5E5%2Bx%5E4-x%5E3-x%5E4-2x%5E3-x%5E2%2Bx%2Bx%5E3-2x%5E2%20%2Bx%20-%201%29)
Factorize the expression
![x^8 - 3x^6 + x^4 - 2x^3 - 1 = (x+1)^2(x^3(x^3-2x^2+x-1) -x(x^3-2x^2+x-1)+1(x^3-2x^2+x-1))](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201%20%3D%20%28x%2B1%29%5E2%28x%5E3%28x%5E3-2x%5E2%2Bx-1%29%20-x%28x%5E3-2x%5E2%2Bx-1%29%2B1%28x%5E3-2x%5E2%2Bx-1%29%29)
Factor out x^3-2x^2+x-1
![x^8 - 3x^6 + x^4 - 2x^3 - 1 = (x+1)^2(x^3 -x+1)(x^3-2x^2+x-1)](https://tex.z-dn.net/?f=x%5E8%20-%203x%5E6%20%2B%20x%5E4%20-%202x%5E3%20-%201%20%3D%20%28x%2B1%29%5E2%28x%5E3%20-x%2B1%29%28x%5E3-2x%5E2%2Bx-1%29)
One of the factors of the above polynomial is
.
This is the same as the option (c)
Hence,
is one of the prime factors of the polynomial
Read more about polynomials at:
brainly.com/question/4142886
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The discount is 12% because 185*0.12=22.2 and to get the discount price you have to subtract the values 185-22.2=162.80.
Answer: ![x^{-4}=\frac{1}{x^{4}}](https://tex.z-dn.net/?f=x%5E%7B-4%7D%3D%5Cfrac%7B1%7D%7Bx%5E%7B4%7D%7D)
Step-by-step explanation:
We have the following expression:
![(\sqrt[3]{x^{2}} \sqrt[6]{x^{4}})^{-3}](https://tex.z-dn.net/?f=%28%5Csqrt%5B3%5D%7Bx%5E%7B2%7D%7D%20%5Csqrt%5B6%5D%7Bx%5E%7B4%7D%7D%29%5E%7B-3%7D)
Which can also be written as:
![(x^{\frac{2}{3}} x^{\frac{4}{6}})^{-3}](https://tex.z-dn.net/?f=%28x%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20x%5E%7B%5Cfrac%7B4%7D%7B6%7D%7D%29%5E%7B-3%7D)
Since we have exponents with the same base
inside the parenthesis, we can sum both exponents:
![(x^{\frac{4}{3}})^{-3}](https://tex.z-dn.net/?f=%28x%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%29%5E%7B-3%7D)
Now, we have to multiply the exponent out of the parenthesis with the inner exponent and have the following result:
Im going to go out on a limb and say its one.