If x^3+x^2+x+a is divisible by x-1 find the value of a
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Answer:
It is given that the polynomial f(x)=x
3
+x
2
−ax+b is divisible by x
2
−x which can be rewritten as x(x−1). It means that the given polynomial is divisible by both x and (x−1) that is they both are factors of f(x)=x
3
+x
2
−ax+b.
Therefore, x=0 and x=1 are the zeroes of f(x) that is both f(0)=0 and f(1)=0.
Let us first substitute x=0 in f(x)=x
3
+x
2
−ax+b as follows:
f(0)=0
3
+0
2
−(a×0)+b
⇒0=0
3
+0
2
−(a×0)+b
⇒0=0+b
⇒b=0
Now, substitute x=1:
f(1)=1
3
+1
2
−(a×1)+b
⇒0=1+1−a+b
⇒0=2−a+b
⇒0=2−a+0(∵b=0)
⇒a=2
Hence, a=2 and b=0.
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