the quotient of $(x^{4}-3x^{2}+4x - 3)$ and $(x^{2}+x - 3)$ is a polynomial. what is the quotient?\n$x^{4}-2x…

Answer:

B. $x^{2}-x + 1$

Explanation:

Step1: Set up long - division

We perform polynomial long - division of $x^{4}-3x^{2}+4x - 3$ by $x^{2}+x - 3$.

Step2: Divide the leading terms

Divide the leading term of the dividend $x^{4}$ by the leading term of the divisor $x^{2}$. $\frac{x^{4}}{x^{2}}=x^{2}$. This is the first term of the quotient.

Step3: Multiply and subtract

Multiply $x^{2}+x - 3$ by $x^{2}$: $x^{2}(x^{2}+x - 3)=x^{4}+x^{3}-3x^{2}$. Subtract this from the dividend: $(x^{4}-3x^{2}+4x - 3)-(x^{4}+x^{3}-3x^{2})=-x^{3}+4x - 3$.

Step4: Repeat the process

Divide the leading term of the new dividend $-x^{3}$ by the leading term of the divisor $x^{2}$. $\frac{-x^{3}}{x^{2}}=-x$. Multiply $x^{2}+x - 3$ by $-x$: $-x(x^{2}+x - 3)=-x^{3}-x^{2}+3x$. Subtract from the new dividend: $(-x^{3}+4x - 3)-(-x^{3}-x^{2}+3x)=x^{2}+x - 3$.

Step5: Final division

Divide the leading term of the new dividend $x^{2}$ by the leading term of the divisor $x^{2}$. $\frac{x^{2}}{x^{2}} = 1$. Multiply $x^{2}+x - 3$ by $1$: $x^{2}+x - 3$. Subtract from the new dividend: $(x^{2}+x - 3)-(x^{2}+x - 3)=0$. So the quotient is $x^{2}-x + 1$.