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.

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)

1^st 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