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I was doing my homework and I am now stuck on question number 7 which is:

The diagram shows the curve with the equation $y = (x + a)(x - b)^2$ where $a$ and $b$ are positive integers.

(i) Write down the values of $a$ and $b$, and also of $c$, given that the curve crosses the $y$-axis at $(0, c)$.

I have attached a picture of my text book here:

I have so far found $b$ like this:

$(x - b)^2 = x^2 - 2bx + b^2$

Since $y = 0$ at $x = 1$,

$0 = 1 - 2b + b^2$

Solving that quadratic equation gives us $b = 1$.

I need to find $a$ and $c$ and I am totally puzzled on how to find them.

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1 Answer

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From the graph it is evident that the roots are $-2$ and $1$ ($1$ being a repeated root),

So the equation becomes: $$(x+2)(x-1)^2=0$$ Comparing it with $$(x+a)(x-b)^2=0$$ We find $a=2,b=1$

Also for finding the point $(0,c)$ we can plug in $x=0$ in the equation $y=(x+2)(x-1)^2$ and we get $c=2$

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