Certainly! Let's walk through the polynomial division step-by-step:
We need to divide [tex]\(2x^2 + x - 3\)[/tex] by [tex]\(x - 1\)[/tex].
1. Set up the division:
[tex]\[
x - 1 \longdiv{ 2x^2 + x - 3 }
\][/tex]
2. First term of the quotient:
To determine the first term of the quotient, divide the leading term of the numerator ([tex]\(2x^2\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]):
[tex]\[
\frac{2x^2}{x} = 2x
\][/tex]
Write [tex]\(2x\)[/tex] as the first term of the quotient above the division bar.
3. Multiply and subtract:
Now, multiply [tex]\(2x\)[/tex] by the entire divisor [tex]\(x - 1\)[/tex]:
[tex]\[
2x \cdot (x - 1) = 2x^2 - 2x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(2x^2 + x - 3) - (2x^2 - 2x) = x + 2x - 3 = 3x - 3
\][/tex]
4. Second term of the quotient:
Now take the leading term in the new polynomial ([tex]\(3x\)[/tex]) and divide it by the leading term of the divisor ([tex]\(x\)[/tex]):
[tex]\[
\frac{3x}{x} = 3
\][/tex]
Write [tex]\(3\)[/tex] as the second term of the quotient.
5. Multiply and subtract:
Multiply [tex]\(3\)[/tex] by the entire divisor [tex]\(x - 1\)[/tex]:
[tex]\[
3 \cdot (x - 1) = 3x - 3
\][/tex]
Subtract this from the current polynomial:
[tex]\[
(3x - 3) - (3x - 3) = 0
\][/tex]
Since the remainder is zero, we have completed our division.
Thus, the quotient is [tex]\(2x + 3\)[/tex] and the remainder is [tex]\(0\)[/tex].
The correct answer to fill in the box is:
[tex]\[
2x + 3
\][/tex]
