Q. 54.7( 6 Votes )
Let us calculate and write the values of a and b if x2 – 4 is a factor of the polynomial ax4 + 2x3 – 3x2 + bx – 4.
Answer :
Formula used.
If f(x) is a polynomial with degree n
Then (x – a) is a factor of f(x) if f(a) = 0
⇒ a2 – b2 = (a + b)(a – b)
1st we find out zero of polynomial g(x)
x2 – 4 = 0
x2 – 22 = (x + 2)(x – 2) = 0
x + 2 = 0 and x – 2 = 0
x = – 2 and x = 2
if x + 2 is factor of f(x) = ax4 + 2x3 – 3x2 + bx – 4
then f( – 2) = 0 ;
f( – 2) = a( – 2)4 + 2( – 2)3 – 3( – 2)2 + b( – 2) – 4
= 16a – 16 – 12 – 2b – 4
16a – 2b – 32 = 0
16a = 32 + 2b ………eq 1
if x – 2 is factor of f(x) = ax4 + 2x3 – 3x2 + bx – 4
then f(2) = 0 ;
f(2) = a(2)4 + 2(2)3 – 3(2)2 + b(2) – 4
= 16a + 16 – 12 + 2b – 4
16a + 2b = 0 ………eq 2
Putting value of 16a from eq 1 into eq 2
(32 + 2b) + 2b = 0
32 + 4b = 0
4b = – 32
b = 
= – 8
Putting value of b in eq 1
16a = 32 + 2 × ( – 8)
16a = 32 – 16
16a = 16
a = 
= 1
Conclusion.
∴ if (x2 – 4) is factor of f(x) then b = – 8 and a = 1
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