%2F(x-2).jpeg "(6x^5-2x^3+4x^2-3x+1)/(x-2)")
Dividing Polynomials: (6x^5-2x^3+4x^2-3x+1)/(x-2)
This article explores the process of dividing the polynomial 6x^5-2x^3+4x^2-3x+1 by the binomial x-2. We will utilize polynomial long division to find the quotient and remainder.
Polynomial Long Division
- Set up the division: Write the dividend (6x^5-2x^3+4x^2-3x+1) inside the division symbol and the divisor (x-2) outside. Notice that we need to include placeholders for missing terms in the dividend (e.g., 0x^4 for the missing x^4 term).
___________
x - 2 | 6x^5 + 0x^4 - 2x^3 + 4x^2 - 3x + 1
- Divide the leading terms: Divide the leading term of the dividend (6x^5) by the leading term of the divisor (x). This gives us 6x^4, which is the first term of the quotient.
6x^4 _______
x - 2 | 6x^5 + 0x^4 - 2x^3 + 4x^2 - 3x + 1
- Multiply the quotient term by the divisor: Multiply 6x^4 by (x-2), which gives us 6x^5 - 12x^4.
6x^4 _______
x - 2 | 6x^5 + 0x^4 - 2x^3 + 4x^2 - 3x + 1
6x^5 - 12x^4
- Subtract: Subtract the result from the dividend. Notice the subtraction changes the signs of the terms.
6x^4 _______
x - 2 | 6x^5 + 0x^4 - 2x^3 + 4x^2 - 3x + 1
6x^5 - 12x^4
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12x^4 - 2x^3
- Bring down the next term: Bring down the next term from the dividend (-2x^3).
6x^4 _______
x - 2 | 6x^5 + 0x^4 - 2x^3 + 4x^2 - 3x + 1
6x^5 - 12x^4
----------------
12x^4 - 2x^3 + 4x^2
- Repeat steps 2-5: Repeat the process of dividing, multiplying, and subtracting until there are no more terms in the dividend to bring down.
6x^4 + 12x^3 + 22x^2 + 48x + 93
x - 2 | 6x^5 + 0x^4 - 2x^3 + 4x^2 - 3x + 1
6x^5 - 12x^4
----------------
12x^4 - 2x^3 + 4x^2
12x^4 - 24x^3
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22x^3 + 4x^2 - 3x
22x^3 - 44x^2
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48x^2 - 3x + 1
48x^2 - 96x
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93x + 1
93x - 186
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187
- The result: The quotient is 6x^4 + 12x^3 + 22x^2 + 48x + 93 and the remainder is 187.
Therefore, the final answer can be expressed as:
(6x^5-2x^3+4x^2-3x+1)/(x-2) = 6x^4 + 12x^3 + 22x^2 + 48x + 93 + 187/(x-2)