Question

The Polynomial. P(x)=x4-2x3+ax+3a-7. when divided by (x+1), leave the remainder 19. find the value of a. also ,find the remainder, when p(x) is divided by (x+2).​

Answer 1

answer-:a=23/2 remainder=43/2

Step-by-step explanation:

x+1=0

x=-1

P(x)=x4-2x3+ax+3a-7=19

P(-1)=(-1)4-2(-1)3+a(-1)+3a-7=19

P(-1)= 1+2-a+3a-7=19

P(-1)=-4+2a=19

P(-1)=2a=19+4

P(-1)=a=23/2

g(x)=x+2

x=-2

P(-2)=(-2)4-2(-2)3+23/2(-2)+23/2(3)-7

P(-2)=16+16-23+39/2-7

P(-2)=(32+32-46+39-14)/2

P(-2)=(64-46+25)/2

P(-2)=18+25/2

P(-2)=43/2

Answer 2

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.