3. Find the value of a, if 2x + 3 is a factor of 6x3 + 19x2 + 13x+a
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Answer:
Let, f(x)=6x
2
+x+a and g(x)=2x−3.
As it is given that g(x) is a factor of p(x),
2x–3=0
⇒2x=3
⇒x=
2
3
Now, by factor theorm we can say that f(
2
3
) is equal to 0,
f(
2
3
)
0
0
0
a
=6(
2
3
)
2
+(
2
3
)+a
=6×
4
9
+
2
3
+a
=
2
27
+
2
3
+a
=15+a
=−15
So,
f(x)
=6x
2
+x−15
=6x
2
−9x+10x−15
=3x(2x−3)+5(2x−3)
=(3x+5)(2x−3)
Hence, value of a is equal to −15 and the factorised from of f(x) is (3x+5)(2x−3).
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