Question

the quotient of $(x^4 - 3x^2 + 4x - 3)$ and $(x^2 + x - 3)$ is a polynomial. what is the quotient?\
\bigcirc $x^4 - 2x^2 + 5x - 6$\
\bigcirc $x^2 - x + 1$\
\bigcirc $x^6 + x^5 - 6x^4 + x^3 + 10x^2 - 15x + 9$\
\bigcirc $x^4 - 4x^2 + 3x$

Explanation:

Step1: Set up polynomial division

We divide $x^4 - 3x^2 + 4x - 3$ by $x^2 + x - 3$. First, divide the leading term of the dividend by the leading term of the divisor:
$\frac{x^4}{x^2} = x^2$
This is the first term of the quotient.

Step2: Multiply divisor by $x^2$

Multiply $x^2 + x - 3$ by $x^2$:
$x^2(x^2 + x - 3) = x^4 + x^3 - 3x^2$

Step3: Subtract from dividend

Subtract the result from the original dividend:
$(x^4 - 3x^2 + 4x - 3) - (x^4 + x^3 - 3x^2) = -x^3 + 4x - 3$

Step4: Next quotient term

Divide the leading term of the new dividend by the leading term of the divisor:
$\frac{-x^3}{x^2} = -x$
This is the second term of the quotient.

Step5: Multiply divisor by $-x$

Multiply $x^2 + x - 3$ by $-x$:
$-x(x^2 + x - 3) = -x^3 - x^2 + 3x$

Step6: Subtract from new dividend

Subtract the result from the current dividend:
$(-x^3 + 4x - 3) - (-x^3 - x^2 + 3x) = x^2 + x - 3$

Step7: Final quotient term

Divide the leading term of the new dividend by the leading term of the divisor:
$\frac{x^2}{x^2} = 1$
This is the third term of the quotient.

Step8: Multiply divisor by 1

Multiply $x^2 + x - 3$ by 1:
$1(x^2 + x - 3) = x^2 + x - 3$

Step9: Subtract to find remainder

Subtract the result from the current dividend:
$(x^2 + x - 3) - (x^2 + x - 3) = 0$
The remainder is 0, so the quotient is the sum of the terms we found.

Answer:

$\boldsymbol{x^2 - x + 1}$ (Option: $x^2 - x + 1$)