$$ y = (x+h)^3(x-h)^2(x-k)(x+k) $$

The equation above is graphed in the \(xy\)-plane. If \(h\) and \(k\) are positive constants and \(h \neq k\), how many distinct \(x\)-intercepts does the graph have?

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The \(x\)-intercepts are the values that cause any of the binomial factors to equal 0.

$$ (x+h)=0 $$ $$ (x-h)=0 $$ $$ (x-k)=0 $$ $$ (x+k)=0 $$ $$ x = -h, h, k, \text{ and } -k $$

Because \(h \neq k\) and the values are positive (not zero), we get \( \boxed{4}\) distinct solutions. The exponents change the multiplicity of the factor, but does not affect the value.