Question

Divide the first polynomial by the binomial and hence find the quotient and remainder.

y⁵ + 5y³ - 8y² + 6y - 16 by y² + 2​

Answer 1

y

5

+5y

3

−8y

2

+6y−16

=y

3

+3y−8

Step-by-step explanation:

\frac{y^5+5y^3-8y^2+6y-16}{y^2+2}

y

2

+2

y

5

+5y

3

−8y

2

+6y−16

Rearrange the terms of numerator of given fraction

\frac{y^5+5y^3+6y-8y^2+16}{y^2+2}

y

2

+2

y

5

+5y

3

+6y−8y

2

+16

\frac{y(y^4+5y^2+6)-8(y^2+2)}{y^2+2}

y

2

+2

y(y

4

+5y

2

+6)−8(y

2

+2)

\frac{y(y^4+2y^2+3y^2+6)-8(y^2+2)}{y^2+2}

y

2

+2

y(y

4

+2y

2

+3y

2

+6)−8(y

2

+2)

\frac{y((y^2(y^2+2)+3(y^2+2))-8(y^2+2)}{y^2+2}

y

2

+2

y((y

2

(y

2

+2)+3(y

2

+2))−8(y

2

+2)

\frac{y(y^2+2)(y^2+3)-8(y^2+2)}{y^2+2}

y

2

+2

y(y

2

+2)(y

2

+3)−8(y

2

+2)

\frac{(y^2+2)(y(y^2+3)-8)}{y^2+2}

y

2

+2

(y

2

+2)(y(y

2

+3)−8)

y(y^2+3)-8y(y

2

+3)−8

y^3+3y-8y

3

+3y−8

Hence,\frac{y^5+5y^3-8y^2+6y-16}{y^2+2}=y^3+3y-8

y

2

+2

y

5

+5y

3

−8y

2

+6y−16

=y

3

+3y−8