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To solve the problem of dividing the polynomial [tex]\(4x^3 + 2x - 1\)[/tex] by the polynomial [tex]\(x + 2\)[/tex], we can use polynomial long division. Here’s a detailed, step-by-step solution:
1. Set up the division:
We arrange the polynomial [tex]\(4x^3 + 2x -1\)[/tex] (the dividend) and [tex]\(x + 2\)[/tex] (the divisor) in a long division format:
[tex]\[ \begin{array}{r|rr} & x + 2 & \\ \hline 4x^3 + 0x^2 + 2x - 1 & & \end{array} \][/tex]
2. Divide the leading terms:
The first term in the dividend is [tex]\(4x^3\)[/tex] and the first term in the divisor is [tex]\(x\)[/tex]. We divide [tex]\(4x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{4x^3}{x} = 4x^2 \][/tex]
This gives us the first term of the quotient [tex]\(4x^2\)[/tex].
3. Multiply and subtract:
Multiply [tex]\(4x^2\)[/tex] by [tex]\(x + 2\)[/tex]:
[tex]\[ 4x^2 \cdot (x + 2) = 4x^3 + 8x^2 \][/tex]
Subtract this from the original polynomial:
[tex]\[ (4x^3 + 0x^2 + 2x - 1) - (4x^3 + 8x^2) = -8x^2 + 2x - 1 \][/tex]
4. Repeat the process:
The next term in the dividend is [tex]\(-8x^2\)[/tex]. Divide [tex]\(-8x^2\)[/tex] by the leading term [tex]\(x\)[/tex]:
[tex]\[ \frac{-8x^2}{x} = -8x \][/tex]
This gives us the next term of the quotient [tex]\(-8x\)[/tex].
Multiply [tex]\(-8x\)[/tex] by [tex]\(x + 2\)[/tex]:
[tex]\[ -8x \cdot (x + 2) = -8x^2 - 16x \][/tex]
Subtract this from the current result:
[tex]\[ (-8x^2 + 2x - 1) - (-8x^2 - 16x) = 18x - 1 \][/tex]
5. Continue the process:
The next term in the dividend is [tex]\(18x\)[/tex]. Divide [tex]\(18x\)[/tex] by the leading term [tex]\(x\)[/tex]:
[tex]\[ \frac{18x}{x} = 18 \][/tex]
This gives us the next term of the quotient [tex]\(18\)[/tex].
Multiply [tex]\(18\)[/tex] by [tex]\(x + 2\)[/tex]:
[tex]\[ 18 \cdot (x + 2) = 18x + 36 \][/tex]
Subtract this from the current result:
[tex]\[ (18x - 1) - (18x + 36) = -37 \][/tex]
6. Combine the results:
At this point, the polynomial [tex]\(-37\)[/tex] has a degree less than [tex]\(x + 2\)[/tex]. So, it is the remainder.
- The quotient is: [tex]\(4x^2 - 8x + 18\)[/tex]
- The remainder is: [tex]\(-37\)[/tex]
Therefore, the result of the division [tex]\( \frac{4x^3 + 2x - 1}{x + 2} \)[/tex] is:
[tex]\[ 4x^2 - 8x + 18 + \frac{-37}{x + 2} \][/tex]
The polynomial division gives us a quotient [tex]\( \left\lfloor \frac{4x^3 + 2x - 1}{x + 2} \right\rfloor = 4x^2 - 8x + 18\)[/tex] and a remainder of [tex]\(\text{Mod}\left(\frac{4x^3 + 2x - 1}{x + 2}\right) = -37\)[/tex].
1. Set up the division:
We arrange the polynomial [tex]\(4x^3 + 2x -1\)[/tex] (the dividend) and [tex]\(x + 2\)[/tex] (the divisor) in a long division format:
[tex]\[ \begin{array}{r|rr} & x + 2 & \\ \hline 4x^3 + 0x^2 + 2x - 1 & & \end{array} \][/tex]
2. Divide the leading terms:
The first term in the dividend is [tex]\(4x^3\)[/tex] and the first term in the divisor is [tex]\(x\)[/tex]. We divide [tex]\(4x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{4x^3}{x} = 4x^2 \][/tex]
This gives us the first term of the quotient [tex]\(4x^2\)[/tex].
3. Multiply and subtract:
Multiply [tex]\(4x^2\)[/tex] by [tex]\(x + 2\)[/tex]:
[tex]\[ 4x^2 \cdot (x + 2) = 4x^3 + 8x^2 \][/tex]
Subtract this from the original polynomial:
[tex]\[ (4x^3 + 0x^2 + 2x - 1) - (4x^3 + 8x^2) = -8x^2 + 2x - 1 \][/tex]
4. Repeat the process:
The next term in the dividend is [tex]\(-8x^2\)[/tex]. Divide [tex]\(-8x^2\)[/tex] by the leading term [tex]\(x\)[/tex]:
[tex]\[ \frac{-8x^2}{x} = -8x \][/tex]
This gives us the next term of the quotient [tex]\(-8x\)[/tex].
Multiply [tex]\(-8x\)[/tex] by [tex]\(x + 2\)[/tex]:
[tex]\[ -8x \cdot (x + 2) = -8x^2 - 16x \][/tex]
Subtract this from the current result:
[tex]\[ (-8x^2 + 2x - 1) - (-8x^2 - 16x) = 18x - 1 \][/tex]
5. Continue the process:
The next term in the dividend is [tex]\(18x\)[/tex]. Divide [tex]\(18x\)[/tex] by the leading term [tex]\(x\)[/tex]:
[tex]\[ \frac{18x}{x} = 18 \][/tex]
This gives us the next term of the quotient [tex]\(18\)[/tex].
Multiply [tex]\(18\)[/tex] by [tex]\(x + 2\)[/tex]:
[tex]\[ 18 \cdot (x + 2) = 18x + 36 \][/tex]
Subtract this from the current result:
[tex]\[ (18x - 1) - (18x + 36) = -37 \][/tex]
6. Combine the results:
At this point, the polynomial [tex]\(-37\)[/tex] has a degree less than [tex]\(x + 2\)[/tex]. So, it is the remainder.
- The quotient is: [tex]\(4x^2 - 8x + 18\)[/tex]
- The remainder is: [tex]\(-37\)[/tex]
Therefore, the result of the division [tex]\( \frac{4x^3 + 2x - 1}{x + 2} \)[/tex] is:
[tex]\[ 4x^2 - 8x + 18 + \frac{-37}{x + 2} \][/tex]
The polynomial division gives us a quotient [tex]\( \left\lfloor \frac{4x^3 + 2x - 1}{x + 2} \right\rfloor = 4x^2 - 8x + 18\)[/tex] and a remainder of [tex]\(\text{Mod}\left(\frac{4x^3 + 2x - 1}{x + 2}\right) = -37\)[/tex].
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