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Polynomial Long Division: (x^2 + x - 17) ÷ (x - 4)
This article will guide you through the process of dividing the polynomial (x^2 + x - 17) by (x - 4) using long division.
Understanding Polynomial Long Division
Polynomial long division is a method for dividing polynomials, similar to the long division you learned in elementary school for numbers. The goal is to find the quotient and remainder of the division.
Step-by-Step Solution:
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Set up the division:
_________ x - 4 | x^2 + x - 17 -
Divide the leading terms:
- x (from the divisor) goes into x^2 (from the dividend) x times.
- Write x above the x^2 term in the quotient.
x x - 4 | x^2 + x - 17 -
Multiply the quotient term by the divisor:
- x * (x - 4) = x^2 - 4x
x x - 4 | x^2 + x - 17 x^2 - 4x -
Subtract the result from the dividend:
- (x^2 + x) - (x^2 - 4x) = 5x
x x - 4 | x^2 + x - 17 x^2 - 4x ------- 5x -
Bring down the next term:
- Bring down the -17 from the dividend.
x x - 4 | x^2 + x - 17 x^2 - 4x ------- 5x - 17 -
Repeat the process:
- x (from the divisor) goes into 5x (from the dividend) 5 times.
- Write 5 next to the x in the quotient.
x + 5 x - 4 | x^2 + x - 17 x^2 - 4x ------- 5x - 17 5x - 20 -
Multiply and subtract:
- 5 * (x - 4) = 5x - 20
- (5x - 17) - (5x - 20) = 3
x + 5 x - 4 | x^2 + x - 17 x^2 - 4x ------- 5x - 17 5x - 20 ------- 3 -
The remainder:
- We are left with a remainder of 3.
Result:
The division of (x^2 + x - 17) by (x - 4) yields a quotient of (x + 5) and a remainder of 3. This can be expressed as:
(x^2 + x - 17) ÷ (x - 4) = x + 5 + 3/(x - 4)
Conclusion:
Polynomial long division is a systematic method for dividing polynomials. By following the steps outlined above, you can successfully find the quotient and remainder of any polynomial division problem.