Question

The polynomial p(y)=ay3+4y2−30y+5 leaves the remainder 5 when divided by q(y)=y+5. Find the value of a?

Answer 1

Answer:

To check that 2y+1 is a multiple p(y) or not, it will be sufficient to check whether 2y+1 is a factor of p(y) or not.

For that,

2y+1=0

⇒y=−

2

1

Now, substitute y=−

2

1

in the polynomial p(y),

p(−

2

1

)=4(−

2

1

)

3

+4(−

2

1

)

2

−(−

2

1

)−1

⇒p(−

2

1

)=4×(−

8

1

)+(4×

4

1

)+

2

1

−1

⇒p(−

2

1

)=−

2

1

+1+

2

1

−1

⇒p(−

2

1

)=0

Here, the remainder is zero 0, when the polynomial p(y)=4y

3

+4y

2

−y−1 is divided by 2y+1.

So, by using the factor theorem now we can say that p(y) is a multiple of 2y+1.