Substitution Question

Question

Substitute $y = x − \frac{a}{3}$ in the equation, $y^3+ay^2+by+c=0$, combine like terms and simplify. What is the coefficient of x?

-I get to where I substitute the first equation into the second, but I do not know how to simplify $$\left(x-\frac{a}{3}\right)^3 + a\left(x-\frac{a}{3}\right)^2 + b\left(x − \frac{a}{3}\right) + c = 0$$ as the exponent and the fraction throw me off. I'm simply looking for the coefficient of x.

Answer 1

we have $$y^3+ay^2+by+c=0$$ plugging $$y=x-\frac{a}{3}$$ in the equation above we get $$(x-\frac{a}{3})^3+a(x-\frac{a}{3})^2+b(x-\frac{a}{3})+c=0$$ expanding and sroting the given terms we obtain $$-1/3\,x{a}^{2}+{\frac {2\,{a}^{3}}{27}}+{x}^{3}+bx-1/3\,ba+c=0$$

Answer 2

Use the binomial formula

$(x+a)^3 = x^3+3x^2a+3xa^2+a^3$

$(x+a)^2 = x^2+2ax + a^2$

and then you see which coefficient is in front of $x$