%2F(x%5E2-3x-2)-long-division.jpeg "(4x^4+5x-4)/(x^2-3x-2) Long Division")
Long Division of Polynomials: (4x^4 + 5x - 4) / (x^2 - 3x - 2)
Long division of polynomials is a method used to divide a polynomial by another polynomial of a lower or equal degree. This process is similar to the long division of numbers. Let's dive into how to perform this division with the example: (4x^4 + 5x - 4) / (x^2 - 3x - 2).
Step 1: Set up the division.
Write the problem in the standard long division format, with the dividend (4x^4 + 5x - 4) inside the division symbol and the divisor (x^2 - 3x - 2) outside.
___________
x^2 - 3x - 2 | 4x^4 + 0x^3 + 0x^2 + 5x - 4
Note: We add 0x^3 and 0x^2 as placeholders to ensure that each term in the dividend is accounted for.
Step 2: Divide the leading terms.
Divide the leading term of the dividend (4x^4) by the leading term of the divisor (x^2). This gives us 4x^2.
4x^2
x^2 - 3x - 2 | 4x^4 + 0x^3 + 0x^2 + 5x - 4
Step 3: Multiply the quotient by the divisor.
Multiply the quotient (4x^2) by the divisor (x^2 - 3x - 2). This gives us 4x^4 - 12x^3 - 8x^2.
4x^2
x^2 - 3x - 2 | 4x^4 + 0x^3 + 0x^2 + 5x - 4
4x^4 - 12x^3 - 8x^2
Step 4: Subtract the result from the dividend.
Subtract the result (4x^4 - 12x^3 - 8x^2) from the dividend (4x^4 + 0x^3 + 0x^2 + 5x - 4). This leaves us with 12x^3 + 8x^2 + 5x - 4.
4x^2
x^2 - 3x - 2 | 4x^4 + 0x^3 + 0x^2 + 5x - 4
4x^4 - 12x^3 - 8x^2
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12x^3 + 8x^2 + 5x - 4
Step 5: Repeat steps 2-4.
Bring down the next term of the dividend (5x). Now, divide the leading term of the new dividend (12x^3) by the leading term of the divisor (x^2). This gives us 12x.
4x^2 + 12x
x^2 - 3x - 2 | 4x^4 + 0x^3 + 0x^2 + 5x - 4
4x^4 - 12x^3 - 8x^2
---------------------
12x^3 + 8x^2 + 5x - 4
12x^3 - 36x^2 - 24x
Subtract the result (12x^3 - 36x^2 - 24x) from the previous result (12x^3 + 8x^2 + 5x - 4). This leaves us with 44x^2 + 29x - 4.
4x^2 + 12x
x^2 - 3x - 2 | 4x^4 + 0x^3 + 0x^2 + 5x - 4
4x^4 - 12x^3 - 8x^2
---------------------
12x^3 + 8x^2 + 5x - 4
12x^3 - 36x^2 - 24x
---------------------
44x^2 + 29x - 4
Step 6: Repeat steps 2-4 again.
Bring down the next term of the dividend (-4). Now, divide the leading term of the new dividend (44x^2) by the leading term of the divisor (x^2). This gives us 44.
4x^2 + 12x + 44
x^2 - 3x - 2 | 4x^4 + 0x^3 + 0x^2 + 5x - 4
4x^4 - 12x^3 - 8x^2
---------------------
12x^3 + 8x^2 + 5x - 4
12x^3 - 36x^2 - 24x
---------------------
44x^2 + 29x - 4
44x^2 - 132x - 88
Subtract the result (44x^2 - 132x - 88) from the previous result (44x^2 + 29x - 4). This leaves us with 161x + 84.
4x^2 + 12x + 44
x^2 - 3x - 2 | 4x^4 + 0x^3 + 0x^2 + 5x - 4
4x^4 - 12x^3 - 8x^2
---------------------
12x^3 + 8x^2 + 5x - 4
12x^3 - 36x^2 - 24x
---------------------
44x^2 + 29x - 4
44x^2 - 132x - 88
---------------------
161x + 84
Step 7: The remainder.
Since the degree of the remainder (161x + 84) is less than the degree of the divisor (x^2 - 3x - 2), we stop here.
Result:
Therefore, the result of the long division is:
(4x^4 + 5x - 4) / (x^2 - 3x - 2) = 4x^2 + 12x + 44 + (161x + 84) / (x^2 - 3x - 2)