Q. \( y^2 = 8x + x^2 + 4x^3 + 4x^4 + 8x^5 \).

Q: What simple algebraic simplification can you do to y^2 = 8x + x^2 + 4x^3 + 4x^4 + 8x^5?

Factor out x: y^2 = x\bigl(8 + x + 4x^2 + 4x^3 + 8x^4\bigr). The quartic factor has no obvious rational roots, so it is likely irreducible over the rationals.

Q: What is the derivative dy/dx by implicit differentiation?

Differentiate: 2y\,dy/dx = 8 + 2x + 12x^2 + 16x^3 + 40x^4. So dy/dx = \dfrac{8 + 2x + 12x^2 + 16x^3 + 40x^4}{2y} = \dfrac{4 + x + 6x^2 + 8x^3 + 20x^4}{y}.

Q: Where are the x-intercepts (real points with y=0)?

Solve 8x + x^2 + 4x^3 + 4x^4 + 8x^5 = 0, i.e. x = 0 or the quartic 8 + x + 4x^2 + 4x^3 + 8x^4 = 0. The quartic’s roots generally require numerical methods; x = 0 is an exact real root.

Q: Is this curve a conic or something else?

This is a hyperelliptic curve given by y^2 = P(x) with \deg P = 5. It is not a conic; over the complex numbers it has genus \left\lfloor\frac{5-1}{2}\right\rfloor = 2..

Q: Are there singular points on the curve?.

Singularities satisfy y=0 and P'(x)=0. So check common roots of P(x)=8x+\cdots+8x^5 and P'(x)=8+2x+12x^2+16x^3+40x^4. Multiple roots of P give singular points; otherwise the curve is smooth.

Q: How do you sketch the real graph quickly?

Plot y=\pm\sqrt{P(x)} where P(x)=8x+x^2+4x^3+4x^4+8x^5. The curve is symmetric about the x-axis. Domain is where P(x)\ge0; near x=0^+, y\approx\pm2\sqrt{2}\,\sqrt{x}..

Q: What is the local behavior near small positive x?.

For small x>0, y \approx \pm\sqrt{8x} = \pm2\sqrt{2}\,\sqrt{x}. Higher-order terms give corrections; for x<0 small, P(x) is typically negative so no real y.

Answer

\[
x^2+4x^3+4x^4=x^2(1+2x)^2,\qquad 8x+8x^5=8x(1+x^4),
\]
so
\[
y^2=x^2(1+2x)^2+8x(1+x^4).
\]
Therefore
\[
y=\pm\sqrt{x^2(1+2x)^2+8x(1+x^4)}.
\]

Detailed Explanation

Write the given relation and note that y is an implicit function of x:
\[y^{2} = 8x + x^{2} + 4x^{3} + 4x^{4} + 8x^{5}\]
Differentiate both sides with respect to x. Use the chain rule on the left side: the derivative of y^{2} is 2y times dy/dx. Differentiate the right side term by term.
\[
\frac{d}{dx}\bigl(y^{2}\bigr) \;=\; \frac{d}{dx}\bigl(8x\bigr) \;+\; \frac{d}{dx}\bigl(x^{2}\bigr) \;+\; \frac{d}{dx}\bigl(4x^{3}\bigr) \;+\; \frac{d}{dx}\bigl(4x^{4}\bigr) \;+\; \frac{d}{dx}\bigl(8x^{5}\bigr)
\]

Compute each derivative:

\[
2y\,\frac{dy}{dx} \;=\; 8 \;+\; 2x \;+\; 12x^{2} \;+\; 16x^{3} \;+\; 40x^{4}
\]
Solve this equation for dy/dx by isolating dy/dx and simplifying (divide numerator and denominator by 2):
\[
\frac{dy}{dx} \;=\; \frac{8 + 2x + 12x^{2} + 16x^{3} + 40x^{4}}{2y}
\;=\; \frac{4 + x + 6x^{2} + 8x^{3} + 20x^{4}}{y}
\]

Note: this expression is valid where y ≠ 0, since division by zero is not allowed.

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Algebra FAQs

What simple algebraic simplification can you do to \(y^2 = 8x + x^2 + 4x^3 + 4x^4 + 8x^5\)?

Factor out \(x\): \(y^2 = x\bigl(8 + x + 4x^2 + 4x^3 + 8x^4\bigr)\). The quartic factor has no obvious rational roots, so it is likely irreducible over the rationals.

What is the derivative \(dy/dx\) by implicit differentiation?

Differentiate: \(2y\,dy/dx = 8 + 2x + 12x^2 + 16x^3 + 40x^4\). So \(dy/dx = \dfrac{8 + 2x + 12x^2 + 16x^3 + 40x^4}{2y} = \dfrac{4 + x + 6x^2 + 8x^3 + 20x^4}{y}\).

Where are the x-intercepts (real points with \(y=0\))?

Solve \(8x + x^2 + 4x^3 + 4x^4 + 8x^5 = 0\), i.e. \(x = 0\) or the quartic \(8 + x + 4x^2 + 4x^3 + 8x^4 = 0\). The quartic’s roots generally require numerical methods; \(x = 0\) is an exact real root.

Is this curve a conic or something else?

This is a hyperelliptic curve given by \(y^2 = P(x)\) with \(\deg P = 5\). It is not a conic; over the complex numbers it has genus \(\left\lfloor\frac{5-1}{2}\right\rfloor = 2\)..

Are there singular points on the curve?.

Singularities satisfy \(y=0\) and \(P'(x)=0\). So check common roots of \(P(x)=8x+\cdots+8x^5\) and \(P'(x)=8+2x+12x^2+16x^3+40x^4\). Multiple roots of \(P\) give singular points; otherwise the curve is smooth.

Does the curve have a vertical tangent at the origin?

Can this curve be parametrized by rational functions?

In general no: a genus-2 hyperelliptic curve does not admit a rational parametrization. Only special lower-genus or reducible cases have rational parametrizations.

How do you sketch the real graph quickly?

Plot \(y=\pm\sqrt{P(x)}\) where \(P(x)=8x+x^2+4x^3+4x^4+8x^5\). The curve is symmetric about the x-axis. Domain is where \(P(x)\ge0\); near \(x=0^+\), \(y\approx\pm2\sqrt{2}\,\sqrt{x}\)..

What is the local behavior near small positive \(x\)?.

For small \(x>0\), \(y \approx \pm\sqrt{8x} = \pm2\sqrt{2}\,\sqrt{x}\). Higher-order terms give corrections; for \(x<0\) small, \(P(x)\) is typically negative so no real \(y\).

Analyze this complex implicit curve
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