Q. 54.7( 6 Votes )

# Let us calculate and write the values of a and b if x^{2} – 4 is a factor of the polynomial ax^{4} + 2x^{3} – 3x^{2} + 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

⇒ a^{2} – b^{2} = (a + b)(a – b)

1^{st} we find out zero of polynomial g(x)

x^{2} – 4 = 0

x^{2} – 2^{2} = (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) = ax^{4} + 2x^{3} – 3x^{2} + 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) = ax^{4} + 2x^{3} – 3x^{2} + 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 (x^{2} – 4) is factor of f(x) then b = – 8 and a = 1

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