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Question 1183420: Pls help me get remainder when 2x^4 - 5x^3 + 7x^2 - x + 6 is divided by x^2+x-2 by using only remainder theorem. Thnx.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source):
Answer by ikleyn(52812)  (Show Source):
You can put this solution on YOUR website! .
Pls help me get remainder when 2x^4 - 5x^3 + 7x^2 - x + 6 is divided by x^2+x-2 by using only remainder theorem.
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@MathLover1 solved the problem, but used the long division of polynomials
instead of the Remainder theorem, as it was requested in the post.
Below I developed the solution to this problem using the Remainder theorem - - - as requested. E N J O Y ( ! )
Since the divisor (x^2 + x - 2) is the quadratic polynomial, the reminder of the division is the linear binomial
and, therefore, has the form (ax+b).
Our task is to find / (to determine) the coefficients "a" and "b" of this linear binomial.
So, we have
2x^4 - 5x^3 + 7x^2 - x + 6 = q(x)*(x^2+x-2) + (ax+b). (1)
where q(x) is another quadratic polynomial, but I even will not touch it . . .
Now notice that the roots of the quadratic polynomial (x^2+x-2) are -2 and 1
(because x^2+x-2 = (x+2)*(x-1) in the factored form).
The remainder of division the quartic polynomial 2x^4 - 5x^3 + 7x^2 - x + 6 by (x+2) is the value of this quartic
at x= -2 (the Remainder theorem), i.e.
2*((-2)^4) - 5*((-2)^3) + 7*((-2)^2) - (-2) + 6 = 108.
Due to equality (1), it means that the linear binomial (ax+b) has the value of 108 at x= -2, i.e.
-2a + b = 108 (2)
The remainder of division the quartic polynomial 2x^4 - 5x^3 + 7x^2 - x + 6 by (x-1) is the value of this quartic
at x= 1 (the Remainder theorem), i.e.
2*(1^4) - 5*(1^3) + 7*(1^2) - 1 + 6 = 9.
Due to equality (1), it means that the linear binomial (ax+b) has the value of 9 at x= 1, i.e.
a + b = 9 (3)
+-------------------------------------------------------------------------+
| Thus we have two equations (2) and (3) to find two unknowns "a" and "b. |
+-------------------------------------------------------------------------+
To solve the system, subtract equation (3) from equation (2) to get
-2a - a = 108-9, -3a = 99, a = 99/(-3) = -33.
Then from (3) you get b= 9 - a = 9 - (-33) = 9 + 33 = 42.
Thus the problem is just solved (by the method you requested), and the ANSWER is:
+---------------------------------------------------------------+
| the remainder of division of 2x^4 - 5x^3 + 7x^2 - x + 6 |
| by (x^2+x-2) is -33x + 42. |
+---------------------------------------------------------------+
Solved as requested and carefully explained.
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