To divide the polynomial [tex]\(6x^3 - 8x + 5\)[/tex] by [tex]\(2x - 4\)[/tex], we follow these steps:
1. Set up the division:
We are given the dividend [tex]\(6x^3 - 8x + 5\)[/tex] and the divisor [tex]\(2x - 4\)[/tex]. We need to perform polynomial long division.
2. First term of the quotient:
- Divide the leading term of the dividend [tex]\(6x^3\)[/tex] by the leading term of the divisor [tex]\(2x\)[/tex]:
[tex]\[
\frac{6x^3}{2x} = 3x^2
\][/tex]
- Multiply [tex]\(3x^2\)[/tex] by the entire divisor [tex]\(2x - 4\)[/tex]:
[tex]\[
3x^2 (2x - 4) = 6x^3 - 12x^2
\][/tex]
3. Subtract the result from the original polynomial:
[tex]\[
(6x^3 - 8x + 5) - (6x^3 - 12x^2) = 12x^2 - 8x + 5
\][/tex]
4. Second term of the quotient:
- Divide the leading term of the new polynomial [tex]\(12x^2\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{12x^2}{2x} = 6x
\][/tex]
- Multiply [tex]\(6x\)[/tex] by the entire divisor [tex]\(2x - 4\)[/tex]:
[tex]\[
6x(2x - 4) = 12x^2 - 24x
\][/tex]
5. Subtract the result from the obtained polynomial:
[tex]\[
(12x^2 - 8x + 5) - (12x^2 - 24x) = 16x + 5
\][/tex]
6. Third term of the quotient:
- Divide the leading term of the new polynomial [tex]\(16x\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{16x}{2x} = 8
\][/tex]
- Multiply [tex]\(8\)[/tex] by the entire divisor [tex]\(2x - 4\)[/tex]:
[tex]\[
8(2x - 4) = 16x - 32
\][/tex]
7. Subtract the result from the obtained polynomial:
[tex]\[
(16x + 5) - (16x - 32) = 37
\][/tex]
8. Result:
The quotient is the sum of the individual terms we have derived: [tex]\(3x^2 + 6x + 8\)[/tex], and the remainder is [tex]\(37\)[/tex].
Therefore,
[tex]\[
\frac{6x^3 - 8x + 5}{2x - 4} = 3x^2 + 6x + 8 + \frac{37}{2x - 4}
\][/tex]
Thus, the quotient is [tex]\(3x^2 + 6x + 8\)[/tex] and the remainder is [tex]\(37\)[/tex].
