Question

hii solve the above question ......​

Answer 1

$\huge\bold{Solution}$

Given :

• p(x) = 8x⁴ + 4x³ - 16x² + 10x + m
• g(x) = 2x-1

Need to find :

• Value of m

Step by step explation :

• p(x) = 8x⁴ + 4x³ - 16x² + 10x+m
• g(x) = 2x-1

➺ 2x-1 = 0

➺ 2x = 1

➺ x = 1/2

Putting value of x in p(x)

p(x) = 8x² + 4x³ - 16x² + 10x+m = 0

➺ 8 x (1/2)⁴ +4x (1/2)³ - 16x (1/2)² + 10×1/2 + m = 0

➺ 8x1/16 + 4×1/8 - 16x1/4 + 10×1/2 + m = 0

➺ 1/2+1/2 - 16/4 + 10/2 + m = 0

${\sf\underline{\purple{Take\:L.C.M :}}}$

➺ (2+2-16+20+4m)/4 = 0

➺ 4-16+20+4m = 0

➺ 24-16+4m = 0

➺ 8+4m = 0

➺ 4m = -8

➺ m = -8/4

➺ m = -2

${\sf\underline{\green{Hence :}}}$

• Value of m = -2

________________________

Answer 2

Question :

Find the value of m for which (2x - 1) is a factor of $({8x}^{4} + {4x}^{3} - {16x}^{2} + 10x + m)$

Answer :

p(x) = 8x⁴ + 4x³ - 16x² + 10x + m

g(x) = 2x-1

⟹ 2x-1 = 0

⟹ 2x = 1

⟹ x = 1/2

Putting value of x in p(x)

p(x) = 8x² + 4x³ - 16x² + 10x + m = 0

⟹ 8 x (1/2)⁴ +4x (1/2)³ - 16x (1/2)² + 10×1/2 + m = 0

⟹ 8x1/16 + 4×1/8 - 16x1/4 + 10×1/2 + m = 0

⟹ 1/2+1/2 - 16/4 + 10/2 + m = 0

Take LCM,

⟹ (2+2-16+20+4m)/4 = 0

⟹ 4 - 16 + 20 + 4m = 0

⟹ 24 - 16 + 4m = 0

⟹ 8 + 4m = 0

⟹ 4m = -8

⟹ m = -8/4

⟹ m = -2

Final Solution :

Value of m for which (2x - 1) is a factor of 8x⁴ + 4x³ - 16x² + 10x + m is (-2).

More Information :

⭐ Factor theorem ➡ x – a is a factor of the polynomial p(x), if p(a) = 0.

Also, if x – a is a factor of p(x), then p(a) = 0, where a is any real number.

Thank you.