The product of two zeroes of the polynomial p(x) = x^3 – 6x^2 + 11x – 6 is 6. Find all the zeroes of
the polynomial.
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Answer:
p(x)=x
3
−6x
2
+11x−6
One of the roots of given polynomial is 3.
p(3)=(3)^3-6(3)^2+11(3)-6 = 0p(3)=(3)
3
−6(3)
2
+11(3)−6=0
We have to find the other two roots.
The polynomial can be factorized as
\begin{gathered}p(x)=x^3-6x^2+11x-6\\\\g(x) = \dfrac{x^3-6x^2+11x-6}{x-3}\\\\g(x) = x^2-3x +2\\g(x) = 0\\\Rightarrow x^2-3x +2= 0\\\Rightarrow x^2 - x -2x + 2 =0\\\Rightarrow x(x-1)-2(x-1)=0\\\Rightarrow (x-1)(x-2) = 0\\\Rightarrow x = 1, x = 2\end{gathered}
p(x)=x
3
−6x
2
+11x−6
g(x)=
x−3
x
3
−6x
2
+11x−6
g(x)=x
2
−3x+2
g(x)=0
⇒x
2
−3x+2=0
⇒x
2
−x−2x+2=0
⇒x(x−1)−2(x−1)=0
⇒(x−1)(x−2)=0
⇒x=1,x=2
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