If f(x)=x^3-(a+b)x^2 +abx, find the value of f(a). What is the significance of x-a?

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Given `f(x)=x^3-(a+b)x^2+abx` :

(a) Determine f(a):

We can factor f(x): first factor out the common x

`f(x)=x(x^2-(a+b)x+ab)` The term in the parantheses factors:

`f(x)=x[(x-a)(x-b)]`

**Now it is clear that f(a)=0.** **f(a)=a(a-a)(a-b)=a*0*(a-b)=0 **

Alternatively you can substitute a for x:

`f(a)=(a)^3-(a+b)(a)^2+ab(a)`

`=a^3-a^3-a^2b+a^2b`

`=0`

(b) What is the significance of (x-a)? **x-a is a linear factor of f(x)**; thus a is a root (or zero or solution) of f(x).

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