Solve 3^10+10!/1!(10-1)!*3^9(-x/5)+10!/2!(10-2)!*3^8(-x/5)^2

$$3 ^ { 10 } + \frac { 10 ! } { 1 ! ( 10 - 1 ) ! } \times 3 ^ { 9 } ( - \frac { x } { 5 } ) + \frac { 10 ! } { 2 ! ( 10 - 2 ) ! } \times 3 ^ { 8 } ( - \frac { x } { 5 } ) ^ { 2 }$$

Evaluate

$$\frac{59049x^{2}}{5}-39366x+59049$$

Short Solution Steps

Calculate 3 to the power of 10 and get 59049.

$$59049+\frac{10!}{1!\left(10-1\right)!}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

The factorial of 10 is 3628800.

$$59049+\frac{3628800}{1!\left(10-1\right)!}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

The factorial of 1 is 1.

$$59049+\frac{3628800}{1\left(10-1\right)!}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Subtract 1 from 10 to get 9.

$$59049+\frac{3628800}{1\times 9!}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

The factorial of 9 is 362880.

$$59049+\frac{3628800}{1\times 362880}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Multiply 1 and 362880 to get 362880.

$$59049+\frac{3628800}{362880}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Divide 3628800 by 362880 to get 10.

$$59049+10\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Calculate 3 to the power of 9 and get 19683.

$$59049+10\times 19683\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Multiply 10 and 19683 to get 196830.

$$59049+196830\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Cancel out 5, the greatest common factor in 196830 and 5.

$$59049-39366x+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

The factorial of 10 is 3628800.

$$59049-39366x+\frac{3628800}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

The factorial of 2 is 2.

$$59049-39366x+\frac{3628800}{2\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Subtract 2 from 10 to get 8.

$$59049-39366x+\frac{3628800}{2\times 8!}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

The factorial of 8 is 40320.

$$59049-39366x+\frac{3628800}{2\times 40320}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Multiply 2 and 40320 to get 80640.

$$59049-39366x+\frac{3628800}{80640}\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Divide 3628800 by 80640 to get 45.

$$59049-39366x+45\times 3^{8}\left(-\frac{x}{5}\right)^{2}$$

Calculate 3 to the power of 8 and get 6561.

$$59049-39366x+45\times 6561\left(-\frac{x}{5}\right)^{2}$$

Multiply 45 and 6561 to get 295245.

$$59049-39366x+295245\left(-\frac{x}{5}\right)^{2}$$

Calculate -\frac{x}{5} to the power of 2 and get \left(\frac{x}{5}\right)^{2}.

$$59049-39366x+295245\times \left(\frac{x}{5}\right)^{2}$$

To raise \frac{x}{5} to a power, raise both numerator and denominator to the power and then divide.

$$59049-39366x+295245\times \frac{x^{2}}{5^{2}}$$

Express 295245\times \frac{x^{2}}{5^{2}} as a single fraction.

$$59049-39366x+\frac{295245x^{2}}{5^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 59049-39366x times \frac{5^{2}}{5^{2}}.

$$\frac{\left(59049-39366x\right)\times 5^{2}}{5^{2}}+\frac{295245x^{2}}{5^{2}}$$

Since \frac{\left(59049-39366x\right)\times 5^{2}}{5^{2}} and \frac{295245x^{2}}{5^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(59049-39366x\right)\times 5^{2}+295245x^{2}}{5^{2}}$$

Do the multiplications in \left(59049-39366x\right)\times 5^{2}+295245x^{2}.

$$\frac{1476225-984150x+295245x^{2}}{5^{2}}$$

Expand 5^{2}.

$$\frac{1476225-984150x+295245x^{2}}{25}$$

Factor

$$\frac{19683\left(3x^{2}-10x+15\right)}{5}$$

Solution Steps

Calculate 3 to the power of 10 and get 59049.

$$factor(59049+\frac{10!}{1!\left(10-1\right)!}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

The factorial of 10 is 3628800.

$$factor(59049+\frac{3628800}{1!\left(10-1\right)!}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

The factorial of 1 is 1.

$$factor(59049+\frac{3628800}{1\left(10-1\right)!}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Subtract 1 from 10 to get 9.

$$factor(59049+\frac{3628800}{1\times 9!}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

The factorial of 9 is 362880.

$$factor(59049+\frac{3628800}{1\times 362880}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Multiply 1 and 362880 to get 362880.

$$factor(59049+\frac{3628800}{362880}\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Divide 3628800 by 362880 to get 10.

$$factor(59049+10\times 3^{9}\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Calculate 3 to the power of 9 and get 19683.

$$factor(59049+10\times 19683\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Multiply 10 and 19683 to get 196830.

$$factor(59049+196830\left(-\frac{x}{5}\right)+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Cancel out 5, the greatest common factor in 196830 and 5.

$$factor(59049-39366x+\frac{10!}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

The factorial of 10 is 3628800.

$$factor(59049-39366x+\frac{3628800}{2!\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

The factorial of 2 is 2.

$$factor(59049-39366x+\frac{3628800}{2\left(10-2\right)!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Subtract 2 from 10 to get 8.

$$factor(59049-39366x+\frac{3628800}{2\times 8!}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

The factorial of 8 is 40320.

$$factor(59049-39366x+\frac{3628800}{2\times 40320}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Multiply 2 and 40320 to get 80640.

$$factor(59049-39366x+\frac{3628800}{80640}\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Divide 3628800 by 80640 to get 45.

$$factor(59049-39366x+45\times 3^{8}\left(-\frac{x}{5}\right)^{2})$$

Calculate 3 to the power of 8 and get 6561.

$$factor(59049-39366x+45\times 6561\left(-\frac{x}{5}\right)^{2})$$

Multiply 45 and 6561 to get 295245.

$$factor(59049-39366x+295245\left(-\frac{x}{5}\right)^{2})$$

Calculate -\frac{x}{5} to the power of 2 and get \left(\frac{x}{5}\right)^{2}.

$$factor(59049-39366x+295245\times \left(\frac{x}{5}\right)^{2})$$

To raise \frac{x}{5} to a power, raise both numerator and denominator to the power and then divide.

$$factor(59049-39366x+295245\times \frac{x^{2}}{5^{2}})$$

Express 295245\times \frac{x^{2}}{5^{2}} as a single fraction.

$$factor(59049-39366x+\frac{295245x^{2}}{5^{2}})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 59049-39366x times \frac{5^{2}}{5^{2}}.

$$factor(\frac{\left(59049-39366x\right)\times 5^{2}}{5^{2}}+\frac{295245x^{2}}{5^{2}})$$

Since \frac{\left(59049-39366x\right)\times 5^{2}}{5^{2}} and \frac{295245x^{2}}{5^{2}} have the same denominator, add them by adding their numerators.

$$factor(\frac{\left(59049-39366x\right)\times 5^{2}+295245x^{2}}{5^{2}})$$

Do the multiplications in \left(59049-39366x\right)\times 5^{2}+295245x^{2}.

$$factor(\frac{1476225-984150x+295245x^{2}}{5^{2}})$$

Calculate 5 to the power of 2 and get 25.

$$factor(\frac{1476225-984150x+295245x^{2}}{25})$$

Consider 1476225-984150x+295245x^{2}. Factor out 98415.

$$98415\left(15-10x+3x^{2}\right)$$

Rewrite the complete factored expression. Simplify. Polynomial 15-10x+3x^{2} is not factored since it does not have any rational roots.

$$\frac{19683\left(15-10x+3x^{2}\right)}{5}$$