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When the polynomial y^3+ay^2+by+1 is divided by y-2 and y+3, the remainder are 9 and 19 respectively, find the value of a and b.​

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Answer:

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When the polynomial y^3+ay^2+by+1 is divided by y-2 and y+3, the remainder are 9 and-example-1
User Fredrik
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7.3k points
4 votes

Answer:

When a polynomial is divided by a linear factor, the remainder is equal to the value of the polynomial at the root of the factor. Therefore, we have:

y - 2 = 0 => y = 2

y + 3 = 0 => y = -3

The remainder when the polynomial is divided by y - 2 is 9, so we have:

(2)^3 + a(2)^2 + b(2) + 1 = 9

8 + 4a + 2b + 1 = 9

4a + 2b = 0

2a + b = 0

Similarly, the remainder when the polynomial is divided by y + 3 is 19, so we have:

(-3)^3 + a(-3)^2 + b(-3) + 1 = 19

-27 + 9a - 3b + 1 = 19

9a - 3b = 45

3a - b = 15

We now have two equations with two variables:

2a + b = 0

3a - b = 15

Solving for b in the first equation and substituting into the second equation, we get:

b = -2a

3a - (-2a) = 15

5a = 15

a = 3

Substituting a = 3 into either of the equations above, we get:

2a + b = 0

2(3) + b = 0

b = -6

Therefore, the values of a and b are a = 3 and b = -6.

User Neetu
by
7.9k points

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