Find the values of a and b if 16x4 - 24x3 + (a-1)x2 + (b+!)x +49 is a perfect square.
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Answer:
It is given that the polynomial 16x
4
−24x
3
+41x
2
−mx+16 to be a perfect square, therefore, we have:
16x
4
−24x
3
+41x
2
−mx+16=(ax
2
+bx+c)
2
⇒16x
4
−24x
3
+41x
2
−mx+16=a
2
x
4
+b
2
x
2
+c
2
+2abx
3
+2cax
2
+2bcx
(∵(a+b+c)
2
=a
2
+b
2
+c
2
+2ab+2bc+2ca)
⇒16x
4
−24x
3
+41x
2
−mx+16=a
2
x
4
+2abx
3
+(2ca+b
2
)x
2
+2bcx+c
2
Comparing the coefficient of x on both sides, we get
a
2
=16⇒a=4
c
2
=16⇒c=4
2ab=−24
⇒2×4×b=−24
⇒8b=−24
⇒b=−
8
24
⇒b=−3
2bc=−m
⇒2×−3×4=−m
⇒−24=−m
⇒m=24
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