Question

The polynomial p(x)=x^4-2x^3+3x^2-ax+3a-7 whwn divided leaves remainder 19.Find value of a.Alao find the remainder,when p(x) is divides by (x+2)​

Answer 1

Answer:

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.

Answer 2

Answer:

p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7

Divisor = x + 1

x + 1 = 0

x = -1

So, substituting the value of x = – 1 in p(x),

we get,

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

Since, a = 5.

We get the polynomial,

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

As per the question,

When the polynomial obtained is divided by (x + 2),

We get, x + 2 = 0

x = – 2

So, substituting the value of x = – 2 in p(x), we get,

p(-2) = (-2)4 – 2(-2)3 + 3(-2)2 – 5(-2) + 8

⇒ p(-2) = 16 + 16 + 12 + 10 + 8

⇒ p(-2) = 62 Therefore, the remainder = 62.