find the quotient and remainder when p X is divided by qx , px= 4+9X 2 minus 4 x square q x is equals to X + 3 x square - 1
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Answer:
Consider the polynomial f(x)=x
6
+x
4
−x
2
−1 and factorise it as follows:
f(x)=x
6
+x
4
−x
2
−1=x
4
(x
2
+1)−1(x
2
+1)=(x
4
−1)(x
2
+1)=(x
2
−1)(x
2
+1)(x
2
+1)
=(x−1)(x+1)(x
2
+1)
2
Therefore, f(x)=(x−1)(x+1)(x
2
+1)
2
Now consider the polynomial g(x)=x
3
−x
2
+x−1 and factorise it as follows:
g(x)=x
3
−x
2
+x−1=x
2
(x−1)+1(x−1)=(x
2
+1)(x−1)
Therefore, g(x)=(x−1)(x
2
+1)
We know that, f(x) = q(x)g(x)+r(x).
Now divide f(x) by g(x) to get q(x):
q(x)=
(x−1)(x
2
+1)
(x−1)(x+1)(x
2
+1)
2
=(x+1)(x
2
+1)=x
3
+x
2
+x+1
Since f(x) is divisible by g(x), therefore, the remainder r(x)=0.
Hence, the quotient q(x)=x
3
+x
2
+x+1 and remainder r(x)=0.
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