Question

Divide 6+19x+x^2-6x^3 by 2+5x-3x^2 and verify the result by Division Algorithm

Answer 1

Answer:Proved

Explanation: Is in attachment

Answer 2

The quotient when $-6x^{3}+x^{2} +19x+6$ divided by $-3x^{2} +5x+2$ is $2x+3$ and the remainder is 0.

Given:

The given divisor is $-3x^{2} +5x+2$

Dividend  is $-6x^{3}+x^{2} +19x+6$

To Find:

We have to divide the dividend with the divisor and obtain the quotient and remainder.

Solution:

To divide the polynomials there are some steps.

Step-1:

we have to divide the highest degree of the dividend with the highest degree of the divisor. The result is the first term in the quotient.

Step-2:

Now multiply the whole divisor with the first term in the quotient and subtract it from the divisor.

Step-3:

The same continues till the remainder becomes the constant.

For better understanding purposes, the division is shown in the attached image.

According to the division algorithm,

a=bq+r

where a is the dividend, b is the divisor, q is the quotient and r is the remainder.

The obtained quotient is 2x+3 and the remainder is 0.

$-6x^{3}+x^{2} +19x+6=(-3x^{2} +5x+2)(2x+3)+0$

$=-6x^{3}+-9x^{2} +10x^{2} +15x+4x+6$

$=-6x^{3}+x^{2} +19x+6$

Therefore, the quotient is 2x+3 and remainder is 0. Also the result is verified by Division Algorithm.

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