Question

If f(x)=x⁴+3x²-ax+b is a polynomial such that it is divided b²y x-1 and x+1 remander are 5 and 19.Find remainder when f(x) is divided my x-2

Answer 1

Answer:-

Given:-

If f(x) = x⁴ + 3x² - ax + b is divided by x - 1 & x + 1 ; the remainders are 5 & 19.

using Remainder Theorem,

We have to substitute the zero of the given linear polynomial in the given biquadratic polynomial.

⟹ x - 1 = 0

⟹ x = 1

Substitute x = 1 in the given polynomial.

⟹ f(1) = 5

⟹ (1)⁴ + 3(1)² - a(1) + b = 5

⟹ 1 + 3(1) - a + b = 5

⟹ b - a = 5 - 1 - 3

⟹ b - a = 1 -- equation (1).

Similarly;

⟹ x + 1 = 0

⟹ x = - 1

Substitute x = - 1 in the given polynomial.

⟹ f( - 1) = 19

⟹ ( - 1)⁴ + 3(- 1)² - a( - 1) + b = 19

⟹ 1 + 3(1) + a + b = 19

⟹ a + b = 19 - 1 - 3

⟹ a + b = 15 -- equation (2).

Add equations (1) & (2).

⟹ b - a + a + b = 1 + 15

⟹ 2b = 16

⟹ b = 16/2

⟹ b = 8

Substitute b = 8 in equation (1).

⟹ 8 - a = 1

⟹ 8 - 1 = a

⟹ 7 = a

Therefore,

By substituting the values of a & b in the polynomial ; the original polynomial will be x⁴ + 3x² - 7x + 8.

Now, we have to find the remainder when the polynomial is divided by x - 2.

zero of x - 2:-

⟹ x - 2 = 0

⟹ x = 2

Substitute x = 2 in the polynomial.

⟹ f(2) = (2)⁴ + 3(2)² - 7(2) + 8

⟹ f(2) = 16 + 3(4) - 14 + 8

⟹ f(2) = 16 + 12 - 6

⟹ f(2) = 22

∴ The remainder obtained when f(x) is divided by x - 2 is 22.

Answer 2

Given :-

• f(x) = x⁴ + 3x² - ax + b
• Divisor = (x - 1), and (x + 1)
• Remainder = 5 , and 19

To Find :-

• The remainder when f(x) divided by (x - 2) = ?

Solution :-

To calculate remainder when f(x) divided by x - 2 , at first we have to set up equation equation then solve the value of a and b after that we have to find the remainder by dividing f(x) by (x - 2). As given in the question that (x - 1) and (x + 1) is the factors of f(x) with different remainder with 5 and 19. Here (x - 1) = 0 , => x = 1. and (x + 1) = 0 => x = - 1

Calculation for 1st equation :-

• R = 5. f(x) = 1 ( Putting x = 1 in f(x))

⇒ f(x) = x⁴ + 3x² - ax + b

⇒ f(1) = (1)⁴ + 3(1)² - a(1) + b

⇒ f(1) = 1 + 3 - a + b

⇒ f(1) = 4 - a + b

⇒ R = 4 - a + b

⇒ a - b = 4 - 5

⇒ a - b = - 1---------(i)

Calculation for 2nd equation :-

• R = 19 f(x) = -1 (Putting x = -1 in f(x))

⇒ f(x) = x⁴ + 3x² - ax + b

⇒ f(-1) = (-1)⁴ + 3(-1)² - a(-1) + b

⇒ f(-1) = 1 + 3 + a + b

⇒ f(-1) = 4 + a + b

⇒ R = 4 + a + b

⇒ a + b = 19 - 4

⇒ a + b = 15-------(ii)

• By solving equations (i) and (ii) we get :-]

⇒ a - b = - 1

⇒ a + b = 15

• By adding equation we get here :-]

⇒ 2a = 14 ⇒a = 7

• Putting the value of a = 7 in (i) :-]

⇒ a - b = -1

⇒ 7 - b = - 1 ⇒- b = - 1 - 7 ⇒ b = 8

• Now calculate remainder here :-

⇒ f(x) = x⁴ + 3x² - ax + b

• Putting a = 7 and b = 8 :-]

⇒ f(x) = x⁴ + 3x² - 7x + 8

• Now dividing f(x) by (x - 2)

x - 2) x⁴ + 3x² - 7x + 8( x³ + 2x² + 7x + 7

x⁴ - 2x³

_____________________

2x³ + 3x² - 7x + 8

2x³ - 4x²

_________________________

7x² - 7x + 8

7x² - 14x

______________________________

7x + 8

7x - 14

___________________________________

22

Therefore , Remainder = 22

• 2nd method by putting x = 2 in f(x) :-]

⇒ f(x) = x⁴ + 3x² - 7x + 8

⇒ f(2) = (2)⁴ + 3(2)² - 7(2) + 8

⇒ f(2) = 16 + 12 - 14 + 8

⇒ f(2) = 36 - 14

⇒ f(2) = 22

Hence,

• Remainder when f(x) dividing by (x - 2) = 22