To find the quotient and the remainder when the polynomial [tex]\(2x^2 - 5x - 4\)[/tex] is divided by [tex]\(x + 2\)[/tex], we can use polynomial long division. Here are the steps involved:
1. Setup the Division:
- Divide [tex]\(2x^2 - 5x - 4\)[/tex] by [tex]\(x + 2\)[/tex].
2. First Division:
- Determine what term multiplied by [tex]\(x + 2\)[/tex] gives us the leading term of [tex]\(2x^2 - 5x - 4\)[/tex].
- The leading term of the dividend is [tex]\(2x^2\)[/tex], and the leading term of the divisor is [tex]\(x\)[/tex]. To match [tex]\(2x^2\)[/tex] with a term from the divisor [tex]\(x + 2\)[/tex], we multiply by [tex]\(2x\)[/tex].
- [tex]\(2x \cdot (x + 2) = 2x^2 + 4x\)[/tex].
3. Subtract:
- Subtract [tex]\(2x^2 + 4x\)[/tex] from [tex]\(2x^2 - 5x - 4\)[/tex]:
[tex]\[
(2x^2 - 5x - 4) - (2x^2 + 4x) = -9x - 4
\][/tex]
4. Second Division:
- Divide the new leading term [tex]\(-9x\)[/tex] by [tex]\(x\)[/tex]:
- [tex]\(-9x / x = -9\)[/tex].
- [tex]\(-9 \cdot (x + 2) = -9x - 18\)[/tex].
5. Subtract Again:
- Subtract [tex]\(-9x - 18\)[/tex] from [tex]\(-9x - 4\)[/tex]:
[tex]\[
(-9x - 4) - (-9x - 18) = -4 + 18 = 14
\][/tex]
6. Conclusion:
- The quotient is the terms we've accumulated from the division steps:
[tex]\[
2x - 9
\][/tex]
- The remainder is the final value we obtained after the last subtraction:
[tex]\[
14
\][/tex]
Therefore, the quotient is [tex]\(2x - 9\)[/tex] and the remainder is [tex]\(14\)[/tex].
To summarize:
When [tex]\(2x^2-5x-4\)[/tex] is divided by [tex]\(x+2\)[/tex]:
- The quotient is [tex]\(2x - 9\)[/tex].
- The remainder is [tex]\(14\)[/tex].
