Question

the polynomial p(x)=x4-2x3+3x2-ax+3a-7 when divided by (x+1) leaves remainder 19.Find the values of a.Also find the remainder when p(x) is divided by (x+2)

Answer 1

Given Polynomial ⇒

P(x) = x⁴ - 2x³ + 3x² - ax + 3a - 7.

Divisor = x + 1

∴ x + 1 = 0

∴ x = -1

Thus,

P(-1) = (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + 3a - 7.

19 = 1 + 2 + 3 + a + 3a - 7

19 = 6 - 7 + 4a

4a - 1 = 19

4a = 20

⇒ a = 5

∴ Value of a is 5.

Now, the Polynomial will be ⇒

P(x) = x⁴ - 2x³ + 3x² - (5)x + 3(5) - 7

P(x) = x⁴ - 2x³ + 3x² - 5x + 15 - 7

P(x) = x⁴ - 2x³ + 3x² - 5x + 8

Now, When this polynomial is divided by (x + 2), then,

x + 2 = 0

x = - 2

∴ P(-2) = (-2)⁴ - 2(-2)³ + 3(-2)² - 5(-2) + 8

⇒ P(-2) = 16 + 16  + 12 + 10 + 8

⇒ P(-2) = 62

Thus, Remainder will be 62.

Hope it helps.

Answer 2

Given:

Polynomial p(x)=x⁴-2x³+3x²-ax+3a-7 which gives remainder 19 when divided by x+1

To Find:

• Value of 'a'.
• Value of remainder when p(x) is divided by x+2

Solution:

Dividend= x⁴-2x³+3x²-ax+3a-7

Divisor= x+1

Remainder= 19

On dividing x⁴-2x³+3x²-ax+3a-7 by x+1, we get

(Calculation in First attachment)

Remainder= 4a-1

Also, it is given that

Remainder=19

⇒ 4a-1= 19

⇒ 4a= 20

⇒ a= 5

Now, after putting value of a in dividend, we get

Dividend= x⁴-2x³+3x²-(5)x+3(5)-7

Dividend= x⁴-2x³+3x²-5x+15-7

Dividend= x⁴-2x³+3x²-5x+8

Now,

Dividend= x⁴-2x³+3x²-5x+8

Divisor= x+2

After dividing x⁴-2x³+3x²-5x+8 by x+2, we get

(Calculation in second attachment)

Remainder= 62

Hence, the value of a is 5 and required remainder is 62.