If the polynomial h(x) = kx 3 − x 2 -xk 3 + 4 is exactly divisible be x-k , then find the value of ‘k.
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Answer:
Let us first divide the given polynomial x
4
+x
3
+8x
2
+ax+b by (x
2
+1) as shown in the above image:
From the division, we observe that the quotient is x
2
+x+7 and the remainder is (a−1)x+(b−7).
Since it is given that x
4
+x
3
+8x
2
+ax+b is exactly divisible by x
2
+1, therefore, the remainder must be equal to 0 that is:
(a−1)x+(b−7)=0
⇒(a−1)x+(b−7)=0⋅x+0
⇒(a−1)=0,(b−7)=0(Bycomparingcoefficients)
⇒a=1,b=7
Hence, a=1 and b=7.
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