Compute It

It is given that $x^3+4x=8$ .

Multiplied by $x^4: \quad x^7+4x^5=8x^4$

$\quad \quad \quad \quad \quad \quad \quad \space \space x^7+4x^2(x^3+4x-4x)=8x(x^3+4x-4x)$

$\quad \quad \quad \quad \quad \quad \quad \space \space x^7+4x^2(8-4x)=8x(8-4x)$

$\quad \quad \quad \quad \quad \quad \quad \space \space x^7+32x^2-16x^3=64x-32x^2$

$\quad \quad \quad \quad \quad \quad \quad \space \space x^7+32x^2-16(8-4x)=64x-32x^2$

$\quad \quad \quad \quad \quad \quad \quad \space \space x^7+32x^2-128+64x=64x-32x^2$

$\quad \quad \quad \quad \quad \quad \quad \space \space x^7+64x^2=\boxed{128}$

${ x }^{ 3 }+4x=8\\$

${ x }^{ 3 }=8-4x$

Squaring both sides.

${ x }^{ 6 }=64+16{ x }^{ 2 }-64x$

${ x }^{ 6 }+64x=16{ x }^{ 2 }+64$

From the given equation:

${ x }^{ 7 }+64{ x }^{ 2 }=x({ x }^{ 6 }+64x)$

${ x }^{ 7 }+64{ x }^{ 2 }=x(16{ x }^{ 2 }+64)=16({ x }^{ 3 }+4x)$

$16(8)=\boxed { 128 }$

let's say dividend = x^7+64x^2 and divider = x^3+4x . Then you will find out by factor theorem that quotient is = x^4 - 4x^2 + 16 and remainder = 64x^2 - 64x . then you can write : x^7+64x^2 = (x^3+4x) (x^4-4x^2+16) + 64x^2-64x ; We know that x^3+4x = 8 ; x (x^2+4) = 8 ; also x = 8/(x^2+4) ; Then Ans = 8 (x^4-4x^2+16) + 64x^2-64x ; Write x^4 - 4x^2 + 16 as (x^2+4)^2 -12x^2 As you know x^2+4 = 8/x ; By doing some substitution like (8-x^3)/x = 4 and interpretation You will get Ans = 32 4 = 128