Complex Calculation

Algebra Level 3

Given a fraction below:

A = 2 x 4 − 21 x 3 + 55 x 2 − 32 x − 4012 x 2 − 10 x + 20 A=\frac{2x^4-21x^3+55x^2-32x-4012}{x^2-10x+20}

Calculate A A when x = 5 − 3 x=5-\sqrt{3} . Type your answer.

Type − 1 -1 if you think A A is undefined.


The answer is 2017.

Tin Le by Tin Le , 15, Vietnam

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1 solution

Tin Le
Aug 19, 2020

x = 5 − 3 ⇔ x − 5 = − 3 x=5-\sqrt{3} \Leftrightarrow x-5 = -\sqrt{3}

⇒ ( x − 5 ) 2 = ( − 3 ) 2 ⇔ x 2 − 10 x + 25 = 3 \Rightarrow (x-5)^2 = (-\sqrt{3})^2 \Leftrightarrow x^2-10x+25 = 3

⇔ x 2 − 10 x + 22 = 0 \Leftrightarrow x^2-10x+22=0

Let's examine the numerator of A A .

2 x 4 − 21 x 3 + 55 x 2 − 32 x − 4012 2x^4-21x^3+55x^2-32x-4012

= ( 2 x 4 − 20 x 3 + 44 x 2 ) − ( x 3 − 10 x 2 + 22 x ) + ( x 2 − 10 x + 22 ) − 4034 = (2x^4-20x^3+44x^2) - (x^3-10x^2+22x) + (x^2-10x+22) - 4034

= 2 x 2 ( x 2 − 10 x + 22 ) − x ( x 2 − 10 x + 22 ) + ( x 2 − 10 x + 22 ) − 4034 = 2x^2(x^2-10x+22) - x(x^2-10x+22) + (x^2-10x+22) - 4034

= ( x 2 − 10 x + 22 ) ( 2 x 2 − x + 1 ) − 4034 =(x^2-10x+22)(2x^2-x+1) - 4034

= − 4034 =-4034 (As x 2 − 10 x + 22 = 0 x^2-10x+22 =0 , as proved above)

Similarly:

x 2 − 10 x + 20 = ( x 2 − 10 x + 22 ) − 2 = − 2 x^2-10x+20 = (x^2-10x+22) - 2 = -2

Substitute the results in A A , we have:

A = − 4034 − 2 = 2017 A=\frac{-4034}{-2} = 2017

Therefore, A = 2017 A=\boxed{2017}

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