Sure! Let's divide [tex]\( 12x^2 - x - 4 \)[/tex] by [tex]\( 4x - 3 \)[/tex] using long division, and find the quotient [tex]\( q(x) \)[/tex] and the remainder [tex]\( r(x) \)[/tex].
1. Setup the Division:
[tex]\[
\begin{array}{r|l}
4x - 3 & 12x^2 - x - 4 \\
\end{array}
\][/tex]
2. Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[
\frac{12x^2}{4x} = 3x
\][/tex]
This gives us the first term of the quotient: [tex]\( 3x \)[/tex].
3. Step 2: Multiply the entire divisor by this term and subtract from the dividend.
[tex]\[
3x \cdot (4x - 3) = 12x^2 - 9x
\][/tex]
Subtract:
[tex]\[
(12x^2 - x - 4) - (12x^2 - 9x) = (12x^2 - x - 4) - 12x^2 + 9x = 8x - 4
\][/tex]
4. Step 3: Bring down the next term from the dividend if necessary.
For this division, we are left with [tex]\( 8x - 4 \)[/tex].
5. Step 4: Repeat the process with the new polynomial [tex]\( 8x - 4 \)[/tex].
[tex]\[
\frac{8x}{4x} = 2
\][/tex]
This gives us the next term of the quotient: [tex]\( 2 \)[/tex].
6. Step 5: Multiply the entire divisor by this new term and subtract from the new dividend.
[tex]\[
2 \cdot (4x - 3) = 8x - 6
\][/tex]
Subtract:
[tex]\[
(8x - 4) - (8x - 6) = (8x - 4) - 8x + 6 = 2
\][/tex]
7. Final Quotient and Remainder:
The division process stops here because the degree of the remainder ([tex]\(2\)[/tex]) is less than the degree of the divisor ([tex]\(4x - 3\)[/tex]).
So, the quotient is:
[tex]\[
q(x) = 3x + 2
\][/tex]
And the remainder is:
[tex]\[
r(x) = 2
\][/tex]
Thus, when we divide [tex]\( 12x^2 - x - 4 \)[/tex] by [tex]\( 4x - 3 \)[/tex], we get
[tex]\[
\left(12 x^2-x-4\right) \div (4 x - 3) = (3x + 2) + \frac{2}{4x - 3}.
\][/tex]
Here, [tex]\( q(x) = 3x + 2 \)[/tex] is the quotient, and [tex]\( r(x) = 2 \)[/tex] is the remainder.
