The polynomial 3 − 3 2 + 4 when divided by ( − 2) leaves as remainder. The polynomial 2 3 − 5 + when divided by ( − 3) leaves as remainder. If − 2 = 16, find the value of .
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Answer:
It is given that a polynomial
ax
3
−
3x
2
+
4
when divided by
(x−
2)
leaves the remainder
p. Let us substitute
x=
2
in
ax
3
− 3x
2
+ 4
p
as follows:
a(2)
3
−3(2)
2
+
4=
p⇒(a×
8)−
(3×
4)+4=p⇒8a−
12+
4=
p
⇒8a−
8=
p....(1)
Similarly, given a polynomial
2x
3
−
5x+
a
when divided by
(x−
2)
leaves the remainder
q. Let us substitute
x=
2
in
2x
3
−
5x+
a
and equate it to
q
as follows:
2(2)
3
− (5× 2)+ a= q ⇒ (2× 8)− 10+ a= q ⇒ 16− 10+ a= q ⇒ 6+ a= q....(2)
It is also given that
p−
2q=
4
, we, now substitute the values of
p and q from equations 1 and 2 as shown below:
p−
2q=
4
⇒
8a−
8−
2(6+
a)=4
⇒
8a−
8−
12−
2a=4
⇒
6a−
20=
4
⇒
6a=
4+
20
⇒
6a=
24
⇒
a=
6
24
⇒
a=
4
Hence,
a=
4
.
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