%2F(x%5E2%2B2x-4).jpeg "(5x^4-2x^3-7x^2-39)/(x^2+2x-4)")
Performing Polynomial Long Division: (5x^4 - 2x^3 - 7x^2 - 39) / (x^2 + 2x - 4)
This article will demonstrate the process of performing polynomial long division on the expression (5x^4 - 2x^3 - 7x^2 - 39) / (x^2 + 2x - 4).
Understanding Polynomial Long Division
Polynomial long division is a method used to divide polynomials, similar to the long division method used for integers. The goal is to find the quotient and remainder of the division.
Steps for Polynomial Long Division
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Set up the problem: Arrange the polynomials in descending order of their exponents, and include placeholders for any missing terms. In this case, the problem is set up as follows:
____________ x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 + 0x - 39 -
Divide the leading terms: Divide the leading term of the dividend (5x^4) by the leading term of the divisor (x^2). This gives us 5x^2. Write this term above the dividend in the quotient space.
5x^2 ________ x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 + 0x - 39 -
Multiply the divisor by the quotient term: Multiply the divisor (x^2 + 2x - 4) by the quotient term (5x^2). This gives us 5x^4 + 10x^3 - 20x^2. Write this result below the dividend.
5x^2 ________ x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 + 0x - 39 5x^4 + 10x^3 - 20x^2 -
Subtract: Subtract the result from the dividend.
5x^2 ________ x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 + 0x - 39 5x^4 + 10x^3 - 20x^2 ------------------ -12x^3 + 13x^2 + 0x -
Bring down the next term: Bring down the next term from the dividend (-39).
5x^2 ________ x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 + 0x - 39 5x^4 + 10x^3 - 20x^2 ------------------ -12x^3 + 13x^2 + 0x - 39 -
Repeat steps 2-5: Repeat the process starting from step 2, using the new dividend (-12x^3 + 13x^2 + 0x - 39). Divide the leading term of the new dividend (-12x^3) by the leading term of the divisor (x^2). This gives us -12x. Write this term in the quotient space.
5x^2 - 12x ______ x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 + 0x - 39 5x^4 + 10x^3 - 20x^2 ------------------ -12x^3 + 13x^2 + 0x - 39 -12x^3 - 24x^2 + 48x -
Subtract and bring down: Subtract the result and bring down the next term.
5x^2 - 12x ______ x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 + 0x - 39 5x^4 + 10x^3 - 20x^2 ------------------ -12x^3 + 13x^2 + 0x - 39 -12x^3 - 24x^2 + 48x ------------------ 37x^2 + 48x - 39 -
Repeat steps 2-5: Repeat the process again. Divide the leading term of the new dividend (37x^2) by the leading term of the divisor (x^2). This gives us 37.
5x^2 - 12x + 37 __ x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 + 0x - 39 5x^4 + 10x^3 - 20x^2 ------------------ -12x^3 + 13x^2 + 0x - 39 -12x^3 - 24x^2 + 48x ------------------ 37x^2 + 48x - 39 37x^2 + 74x - 148 -
Subtract: Subtract the result.
5x^2 - 12x + 37 __ x^2 + 2x - 4 | 5x^4 - 2x^3 - 7x^2 + 0x - 39 5x^4 + 10x^3 - 20x^2 ------------------ -12x^3 + 13x^2 + 0x - 39 -12x^3 - 24x^2 + 48x ------------------ 37x^2 + 48x - 39 37x^2 + 74x - 148 ------------------ -26x + 109
The Result
The quotient of the division is 5x^2 - 12x + 37 and the remainder is -26x + 109. Therefore, we can express the original expression as:
(5x^4 - 2x^3 - 7x^2 - 39) / (x^2 + 2x - 4) = 5x^2 - 12x + 37 + (-26x + 109)/(x^2 + 2x - 4)