%2F(x-1).jpeg "(3x^3-5x^2-2)/(x-1)")
Polynomial Long Division: (3x^3 - 5x^2 - 2) / (x - 1)
Polynomial long division is a method used to divide polynomials. It's similar to the long division you learned in elementary school, but with variables and exponents. Let's break down the division of (3x^3 - 5x^2 - 2) by (x - 1).
Steps:
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Set up the division:
- Write the dividend (3x^3 - 5x^2 - 2) inside the division symbol.
- Write the divisor (x - 1) outside the division symbol.
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Divide the leading terms:
- Divide the leading term of the dividend (3x^3) by the leading term of the divisor (x). This gives us 3x^2.
- Write 3x^2 above the division symbol, aligned with the x^2 term of the dividend.
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Multiply the divisor by the quotient term:
- Multiply the entire divisor (x - 1) by the quotient term we just found (3x^2). This gives us 3x^3 - 3x^2.
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Subtract:
- Subtract the result from the dividend. This leaves us with -2x^2 - 2.
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Bring down the next term:
- Bring down the next term from the dividend (-2).
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Repeat steps 2-5:
- Divide the leading term of the new dividend (-2x^2) by the leading term of the divisor (x). This gives us -2x.
- Write -2x above the division symbol, aligned with the x term.
- Multiply the divisor (x - 1) by -2x. This gives us -2x^2 + 2x.
- Subtract this result from the current dividend. This leaves us with -2x - 2.
- Bring down the next term (which is the constant term -2).
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Repeat steps 2-5 again:
- Divide the leading term of the new dividend (-2x) by the leading term of the divisor (x). This gives us -2.
- Write -2 above the division symbol.
- Multiply the divisor (x - 1) by -2. This gives us -2x + 2.
- Subtract this result from the current dividend. This leaves us with -4.
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Remainder:
- The last result, -4, is our remainder.
Result:
Therefore, the result of dividing (3x^3 - 5x^2 - 2) by (x - 1) is:
3x^2 - 2x - 2 with a remainder of -4
We can express this as:
(3x^3 - 5x^2 - 2) / (x - 1) = 3x^2 - 2x - 2 - 4/(x - 1)