Determine the values of a M and N , so that the polynomial 2x^3+mx^2+nx-3 and X^3-3mx^2+2nx+4 are both divisible by x-2.
I am so lost, i do not know how to start this problem! please help i have a test soon!
Continuing from the partial solution from 10 years ago, except I've fixed the mistake in getting to one of the equations.
(1) -6m + 2n + 6
(2) 4m + 2n + 13
Proceed with elimination:
-6m + 2n + 6 = 4m + 2n + 13
-6m + 6 = 4m + 13
10m = -7
m = -0.7
Proceed with substitution:
(2) 4m + 2n + 13
4(-0.7) + 2n + 13 = 0
-2.8 + 2n + 13 = 0
2n = -10.2
n = -5.1
Hopefully this goes through the solution well enough.
(Also, it's pretty crazy that the first part of this solution was done over 10 years ago... collaboration across time!!!)
So what is the answer?
How are you getting -6m + 2n + 6 as the one equation? I keep on getting -12m-4n+12. Am I missing a step or just doing it wrong?
Well, don't worry, I'm here to help you with a smile! Let's break it down step by step.
To determine if a polynomial is divisible by another polynomial, we need to check if the remainder is zero when we divide them. In this case, we need to check if the remainder is zero when dividing both polynomials by x-2.
Let's start with the first polynomial, 2x^3 + mx^2 + nx - 3.
When we divide this polynomial by x-2, we use synthetic division. The process involves substituting 2 into the polynomial and performing the division.
Using synthetic division, we have:
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2 | 2 m n -3
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First, bring down the leading coefficient, which is 2.
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2 | 2 m n -3
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4
Next, multiply 2 by 2 and write the result in the next column.
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2 | 2 m n -3
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4 4(m)
Add the two values in the second column to get the third column.
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2 | 2 m n -3
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4 4(m) 4(m) + n
Multiply 2 by the last number in the third column.
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2 | 2 m n -3
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4 4(m) 4(m) + n 2(4(m) + n)
Add the two values in the third column to get the fourth column.
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2 | 2 m n -3
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4 4(m) 4(m) + n 2(4(m) + n) - 3
Now, if the remainder is zero, then the polynomial is divisible by x-2. In this case, the remainder is -3 + 2(4m + n) - 3, which simplifies to:
-6 + 8m + 2n
To make this remainder zero, we need -6 + 8m + 2n = 0.
So, one equation we have is: -6 + 8m + 2n = 0.
Now let's move on to the second polynomial, x^3 - 3mx^2 + 2nx + 4.
Using similar synthetic division, we have:
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2 | 1 -3m 2n 4
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2 2(2)
4 4m+4n
8m+8n 8m+8n
16m+16n + 4
The remainder for this polynomial is: 16m + 16n + 4.
To make this remainder zero, we need 16m + 16n + 4 = 0.
So, the second equation we have is: 16m + 16n + 4 = 0.
Now, you have two equations:
-6 + 8m + 2n = 0 --> Equation 1
16m + 16n + 4 = 0 --> Equation 2
You can solve these equations simultaneously to find the values of m and n that satisfy both equations, and voila!
To determine the values of a and b, we need to use the fact that if a polynomial p(x) is divisible by a factor (x - c), then p(c) = 0.
First, let's consider the polynomial 2x^3 + mx^2 + nx - 3. Since it is divisible by (x - 2), we know that p(2) = 0.
To find the value of m and n, substitute x = 2 into the polynomial:
2(2)^3 + m(2)^2 + n(2) - 3 = 0.
Simplifying this equation, we get:
16 + 4m + 2n - 3 = 0.
19 + 4m + 2n = 0.
This gives us the equation 4m + 2n = -19. (Equation 1)
Now, let's consider the polynomial x^3 - 3mx^2 + 2nx + 4. Since it is also divisible by (x - 2), we know that p(2) = 0.
Substituting x = 2 into the polynomial:
2^3 - 3m(2)^2 + 2n(2) + 4 = 0.
Simplifying this equation, we get:
8 - 12m + 4n + 4 = 0.
16 - 12m + 4n = 0.
Dividing this equation by 4, we obtain:
4 - 3m + n = 0.
This gives us the equation -3m + n = -4. (Equation 2)
Now, we have two equations:
4m + 2n = -19 (Equation 1)
-3m + n = -4 (Equation 2)
To find the values of m and n, we can solve this system of equations by elimination or substitution.
To solve by elimination, multiply Equation 1 by 3 and multiply Equation 2 by 4 to make the coefficients of "m" equal:
12m + 6n = -57 (Equation 3)
-12m + 4n = -16 (Equation 4)
Now, add Equation 3 and Equation 4:
12m + 6n - 12m + 4n = -57 - 16
10n = -73
n = -73/10
Now substitute the value of n into Equation 2 to find m:
-3m + (-73/10) = -4
-3m = -4 + (73/10)
-3m = (-40 + 73)/10
-3m = 33/10
m = -11/10
So, the values of m and n that make both polynomials divisible by (x - 2) are m = -11/10 and n = -73/10.
Just change the variable x with the value x = 2. So, we get:
f(x) = 2x^3 + mx^2 + nx - 3
f(2) = 0 = 2(8) + 4m + 2n - 3
0 = 13 + 4m + 2n. ..(1)
g(x) = 3mx^2 + 2nx + 4
g(2) = 0 = 12m + 4n + 4
0 = 6m + 2n + 2. ..(2)
Using elimination and substitution for both equation (1) and (2), you'll get the values of m and n