Find the values of m and n if y^4 + my^3 + 2y^2- 3y + n is divisible by y^2-1
To find the values of m and n such that the polynomial y^4 + my^3 + 2y^2 - 3y + n is divisible by y^2 - 1, we can use polynomial division.
Step 1: Set up the polynomial division:
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y^2 - 1 | y^4 + my^3 + 2y^2 - 3y + n
We need to divide y^4 + my^3 + 2y^2 - 3y + n by y^2 - 1.
Step 2: Divide the first term y^4 by y^2.
y^4 / y^2 = y^2
Step 3: Multiply the divisor y^2 - 1 by the quotient y^2.
(y^2 - 1) * (y^2) = y^4 - y^2
Step 4: Subtract the product obtained in step 3 from the original polynomial.
(y^4 + my^3 + 2y^2 - 3y + n) - (y^4 - y^2) = my^3 + 3y^2 - 3y + n + y^2
Step 5: Continue the division with the new polynomial obtained in step 4.
Step 6: Divide the first term in my^3 by y^2.
(my^3) / (y^2) = m * y
Step 7: Multiply the divisor y^2 - 1 by the quotient m * y.
(y^2 - 1) * (m * y) = m * y^3 - m * y
Step 8: Subtract the product obtained in step 7 from the polynomial obtained in step 4.
(my^3 + 3y^2 - 3y + n + y^2) - (m * y^3 - m * y) = 4y^2 -3y + n + m * y
Step 9: Continue the division with the new polynomial obtained in step 8.
Step 10: Divide the first term in 4y^2 by y^2.
(4y^2) / (y^2) = 4
Step 11: Multiply the divisor y^2 - 1 by the quotient 4.
(y^2 - 1) * 4 = 4y^2 - 4
Step 12: Subtract the product obtained in step 11 from the polynomial obtained in step 8.
(4y^2 -3y + n + m * y) - (4y^2 - 4) = -3y + n + m * y + 4
Step 13: Since we are dividing by y^2 - 1, the remainder should be zero.
Therefore, -3y + n + m * y + 4 = 0
Step 14: Simplify the equation obtained in step 13.
-3y + m * y + n + 4 = 0
Step 15: Combine like terms.
(m - 3)y + (n + 4) = 0
For this equation to hold true for all values of y, the coefficient of y, (m - 3), must be zero.
Therefore, m - 3 = 0
Solving for m, we get:
m = 3
Step 16: Substitute the value of m into the equation obtained in step 13.
-3y + n + 3y + 4 = 0
Step 17: Combine like terms.
n + 4 = 0
Step 18: Solve for n.
n = -4
Therefore, the values of m and n that make the polynomial y^4 + my^3 + 2y^2 - 3y + n divisible by y^2 - 1 are:
m = 3
n = -4
To find the values of m and n for which the polynomial y^4 + my^3 + 2y^2 - 3y + n is divisible by y^2 - 1, we need to perform polynomial division.
First, let's divide the polynomial y^4 + my^3 + 2y^2 - 3y + n by y^2 - 1:
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y^2 - 1 | y^4 + my^3 + 2y^2 - 3y + n
To start the division, we divide y^4 by y^2, which gives us y^2. We then multiply y^2 by y^2 - 1 to get y^4 - y^2. Subtracting this result from the original polynomial, we get:
y^2 - 1
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y^2 - 1 | y^4 + my^3 + 2y^2 - 3y + n
- (y^4 - y^2)
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my^3 + 3y^2 - 3y + n
We then repeat the process, dividing my^3 by y^2, which gives us my. We multiply my by y^2 - 1 to get my^3 - my. We subtract this result from the previous result:
y^2 - 1 + my
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y^2 - 1 | y^4 + my^3 + 2y^2 - 3y + n
- (y^4 - y^2)
----------------
my^3 + 3y^2 - 3y + n
- (my^3 - my)
-----------------
3y^2 - 3y + n + my
We repeat this process one more time, dividing 3y^2 by y^2, which gives us 3. We multiply 3 by y^2 - 1 to get 3y^2 - 3. We subtract this result from the previous result:
y^2 - 1 + my + 3
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y^2 - 1 | y^4 + my^3 + 2y^2 - 3y + n
- (y^4 - y^2)
----------------
my^3 + 3y^2 - 3y + n
- (my^3 - my)
-----------------
3y^2 - 3y + n + my
- (3y^2 - 3)
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n + my + 3
At this point, we have no more terms to divide, but we still have the remainder n + my + 3. For the given polynomial to be divisible by y^2 - 1, the remainder must be equal to 0.
Therefore, we can set the remainder equal to 0:
n + my + 3 = 0
This equation allows us to solve for the values of m and n.
Find the value of m and n if y2 - 1 is a factor of y4 + my + 2y2 - 3y + n.
Just do the division. The remainder is
(m-3)y + (n+3)
To be divisible, the remainder must be zero, so that means
m=3 and n = -3.
Check:
y^4 + 3y^3 + 2y^2 - 3y - 3 = (y^2-1)(y^2+3y+3)