mx3-nx2+5x-1 is divided by x-2 the remainder is -39. When divided by x-1 the remainder is 3. What are the values of m and n. Didn't know how to do the superscript for mx3
use ^ for exponents
using synthetic division, the remainders are:
(x-2): 8m-4n+9
(x-1): m-n+4
so, we now know
8m-4n+9 = -39
m-n+4 = 3
(m,n) = (-11,-10)
f(x) = -11x^2 + 10x^2 + 5x - 1
To find the values of m and n, we can use polynomial long division.
First, let's divide mx^3 - nx^2 + 5x - 1 by x - 2.
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x - 2 | mx^3 - nx^2 + 5x - 1
-(mx^3 - 2nx^2)
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nx^2 + 5x - 1
-(nx^2 - 2n x)
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7x - 1
-(7x - 14)
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13
According to the given information, the remainder when divided by x - 2 is -39. Since the remainder we obtained is 13, this implies that 13 = -39.
However, -39 ≠ 13, which means there is no real solution for m and n that satisfies the given conditions.
To find the values of m and n, we can use the remainder theorem.
According to the remainder theorem, if we divide a polynomial f(x) by x - a and the remainder is R, then f(a) = R.
First, let's divide mx^3 - nx^2 + 5x - 1 by x - 2 and set the remainder equal to -39:
mx^3 - nx^2 + 5x - 1 = (x - 2)Q(x) - 39
Where Q(x) is the quotient obtained when dividing mx^3 - nx^2 + 5x - 1 by x - 2.
Next, let's divide the same polynomial by x - 1 and set the remainder equal to 3:
mx^3 - nx^2 + 5x - 1 = (x - 1)P(x) + 3
Where P(x) is the quotient obtained when dividing mx^3 - nx^2 + 5x - 1 by x - 1.
Now, we can substitute x = 2 into the equation mx^3 - nx^2 + 5x - 1 = (x - 2)Q(x) - 39:
m(2)^3 - n(2)^2 + 5(2) - 1 = (2 - 2)Q(2) - 39
8m - 4n + 10 - 1 = 0 - 39
8m - 4n + 9 = -39
8m - 4n = -48
Similarly, substituting x = 1 into the equation mx^3 - nx^2 + 5x - 1 = (x - 1)P(x) + 3:
m(1)^3 - n(1)^2 + 5(1) - 1 = (1 - 1)P(1) + 3
m - n + 5 - 1 = 0 + 3
m - n + 4 = 3
m - n = -1
Now we have a system of equations:
8m - 4n = -48
m - n = -1
We can solve this system by substitution or elimination. Multiplying the second equation by 4, we have:
4m - 4n = -4
Adding this equation to the first equation:
8m - 4n + 4m - 4n = -48 - 4
12m - 8n = -52
Dividing both sides by 4:
3m - 2n = -13
Now we have a new equation:
3m - 2n = -13
Let's multiply the second equation by 2:
2m - 2n = -2
Adding this equation to the first equation:
3m - 2n + 2m - 2n = -13 - 2
5m - 4n = -15
Dividing both sides by 5:
m - (4/5)n = -3
From the equation above, we see that m must be equal to -3. Now we can substitute m = -3 into the second equation:
-3 - n = -1
Adding n to both sides:
-3 = 1 - n
Adding 3 to both sides:
0 = 4 - n
Subtracting 4 from both sides:
n = 4
Therefore, the values of m and n are -3 and 4, respectively.