Question

divide polynomial p(x)=3x^4+5x^3-7x^2+2x+3 by q(x)=x^2+3x+1 what should be subtracted from the polynomial p(x) so that it is divisible by q(x).

Answer 1

1 should be subtracted for p(x) so that it is divisible by q(x)

Answer 2

Given:

The dividend, p(x) = $3x^{4} + 5x^3 - 7x^2 + 2x + 3$.

The divisor, q(x) = $x^2 + 3x + 1$.

To Find:

We have to find out what should be subtracted from the polynomial p(x) so that it is divisible by q(x).

Solution:

p(x) = $3x^{4} + 5x^3 - 7x^2 + 2x + 3$.

q(x) = $x^2 + 3x + 1$.

The steps of division are given below.

$\frac{p(x)}{q(x)} = \frac{3x^4+5x^3-7x^2+2x+3}{x^2+3x+1}$

On dividing $3x^{4} + 5x^3 - 7x^2 + 2x + 3$ by $x^2 + 3x + 1$,

The quotient = $3x^2$.

The reminder = $-4x^3 - 10x^2 + 2x$.

Now, on dividing the $-4x^3 - 10x^2 + 2x$ by $x^2 + 3x + 1$ we get,

The quotient = $3x^2-4x$.

The reminder = $2x^2 + 6x +3$.

Dividing $2x^2 + 6x +3$ by $x^2 + 3x + 1$ we get,

The quotient = $3x^2-4x +2$.

The reminder = $1$.

Hence, the number that should be subtracted from the polynomial p(x) so that it is divisible by q(x) is 1.

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