Finding a constant in polynomial long division
- Thread starter thunc14
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HallsofIvy
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Did you not try just doing the division?
Otis
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Hi thunc. We begin by completing the polynomial long-division because that's how we find the remainder. (If you're not familiar with dividing polynomials long-hand, you can google the topic for lessons and worked exercises.)
Here's a similar example, to explain what's intended by stating a constant remainder:
(3x^4 - kx^2 + x - 5) ÷ (x^2 - x + 1)
Polynomial long-division gives us the quotient and remainder.
Quotient: 3x^2 + 3x - k
Remainder: k - 5 - (k + 2)x
What does it mean to say that the remainder is constant? It means the remainder cannot contain the variable x (because different values of x would yield different remainders instead of a single value).
There's only one way the expression k - 5 - (k + 2)x can be constant, and that's if k+2 is zero. In other words, the coefficient on x must be zero (to rid the remainder of its variable term).
k + 2 = 0
Therefore, the remainder is constant when k is -2.
Follow the same reasoning, in your exercise. If you need more help, please show us the results of your polynomial long-division and the work you tried after that.
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Thank you for the help. I arrived at the same answer for your example. For my problem, I divided it all out, and unless my arithmetic is wrong, I got that k should be (-37/9) to make x equal to zero for the remainder. That is not an answer choice, so perhaps I made an arithmetic error somewhere?
