Question

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Answer 1

$\sf{\large{\red{\underline{Question:-}}}}$

The polynomial p(x) = x⁴ - 2x³ + 3x² - ax + b when divided by (x - 1) and (x + 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence find the remainder when p(x) is divided by (x - 2)

$\large{\sf{\red{\underline{Formulas\: used:-}}}}$

• Remainder theorem
• Linear equation in double variable

$\large{\sf{\red{\underline{Answer:-}}}}$

Remainder Theorem:

When p(x) is divided by (x - a), then p(a) is the remainder.

Here,

p(x) = x⁴ - 2x³ + 3x² - ax + b

Dividing by (x - 1)

Here, p(x) is divided by (x - 1) $\implies$a is 1. Hence remainder is p(1)

Value of p(1)

p(1) = 1⁴ - (2 * 1³) + (3 * 1²) - (a * 1) + b

$\implies$ p(1) = 1 - 2 + 3 - a + b

$\implies$ p(1) = 2 + b - a --------->Remainder

It is given that remainder is 5. So,

2 + b - a = 5

$\implies$ $\large{\mathbb{b\:-\:a\:=\:3}}$ ---------(1)

Dividing by (x - 1)

Here, p(x) is divided by {x - (-1) } $\implies$a is -1. Hence remainder is p(-1)

Value of p(-1)

p(-1) = (-1)⁴ - (2 * -1)³ + (3 * -1)² - (a * -1) + b

p(-1) = 1 + 2 + 3 + a + b

p(-1) = 6 + a + b --------->Remainder

It is given that the value of remainder is 19

6 + a + b = 19

$\implies$ $\large{\mathbb{a\:+\:b\:=\:3}}$ --------(2)

• b - a = 3
• b + a = 13

Adding (1) and (2)

Finding b:

b - a +b + a = 3 + 13

$\implies$ 2b = 16

$\implies$ b = 16/2 = 8

Finding a:

b - a = 3

$\implies$ 8 - a = 3

$\implies$ a = 5

$\large{\red{\bf{a\:=\:5}}}$

$\large{\red{\bf{b\:=\:8}}}$

p(x) = x⁴ - 2x³ + 3x² - 5x + 8

Remainder when p(x) is divided by (x - 2)

Here, p(x) is divided by {x - 2 } $\implies$a is 2. Hence remainder is p(2)

p(2) = 2⁴- (2 * 2³) + (3 * 2²) - (5 * 2) + 8

$\implies$ p(2) = 16 - 16 + 12 - 10 + 8

$\implies$ p(2) = 10

$\large{\red{\bf{Remainder\:=\:10}}}$

Answer 2

Answer:

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