if a polynomial leaves a remainder of 5 when divided by x-3 and a reminder of -7 when divided by x+1, what is the reminder when the polynomial is divided by x^2-2x-3?
Let's say that p(x) = (x-3)(x+1)q(x) + (ax+b)
That is, it has some linear remainder when divided by x^2-2x-3
We know from the Remainder Theorem that
p(3) = 5
p(-1) = -7
That means that
3a+b = 5
-a+b = -7
So, a=3, b=-4
The desired remainder is 3x-4
To find the remainder when a polynomial is divided by another polynomial, we can use polynomial long division. Here's how you can find the remainder in this case:
1. Begin by expressing the given polynomial as a general polynomial divided by the divisor polynomial:
Dividend = (x^2 - 2x - 3) * Quotient + Remainder
2. Set up the long division, where the divisor is x^2 - 2x - 3 and the dividend is the general polynomial:
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x^2 - 2x - 3 | General Polynomial
3. Divide the first term of the general polynomial (highest degree) by the first term of the divisor (x^2). This will give you the first term of the quotient.
4. Multiply the divisor (x^2 - 2x - 3) by the first term of the quotient and subtract the result from the general polynomial. Write down the remainder after this subtraction and bring down the next term of the general polynomial.
5. Repeat steps 3 and 4 until you have subtracted all the terms of the general polynomial.
6. The remainder obtained is your final answer.
Let's work through the division:
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x^2 - 2x - 3 | General Polynomial
To find the first term of the quotient, divide the first term of the general polynomial by the first term of the divisor, which is (x^2). Let's assume the first term of the quotient is 'a':
a * (x^2 - 2x - 3)
The first term of the general polynomial may be represented as 'k1 * x^2', where 'k1' is a constant. Thus:
k1 * x^2 / (x^2) = a
k1 = a
Subtracting 'a * (x^2 - 2x - 3)' from the general polynomial gives us the remainder:
General Polynomial - a * (x^2 - 2x - 3) = Remainder1
Now, bring down the next term (-2x). We can repeat the process:
Remainder1 - b * (x^2 - 2x - 3) = Remainder2
After extracting the remainder, we can divide the remainder by the first term of the divisor 'x^2 - 2x - 3' again, and continue this process until we have subtracted all the terms.
Finally, the last remainder obtained will be the answer to your original question, which is the remainder when the polynomial is divided by x^2 - 2x - 3.
To find the remainder when a polynomial is divided by another polynomial, we can use the remainder theorem. According to the theorem, if a polynomial f(x) leaves a remainder r when divided by another polynomial g(x), then f(c) = r, where c is the root of g(x).
Given that the polynomial leaves a remainder of 5 when divided by (x-3) and a remainder of -7 when divided by (x+1), we can consider the roots of those divisors.
x - 3 = 0
x = 3
x + 1 = 0
x = -1
So, the roots are x = 3 and x = -1.
Now, we need to find the remainder when the polynomial is divided by x^2 - 2x - 3.
x^2 - 2x - 3 = (x - 3)(x + 1)
Since the remainder theorem states that f(c) = r, where c is the root of g(x), we substitute x = 3 into the polynomial to find the remainder when divided by (x - 3):
f(3) = 5
Similarly, we substitute x = -1 into the polynomial to find the remainder when divided by (x + 1):
f(-1) = -7
Now, we can set up a system of equations to determine the coefficients of the polynomial. Let's assume the polynomial is represented by f(x) = ax^2 + bx + c.
f(3) = 5 => 9a + 3b + c = 5 ----(1)
f(-1) = -7 => a - b + c = -7 ----(2)
Solving these equations simultaneously, we get:
a = -1
b = 3
c = 4
Therefore, the polynomial is f(x) = -x^2 + 3x + 4.
Finally, to find the remainder when f(x) = -x^2 + 3x + 4 is divided by x^2 - 2x - 3, we divide them:
(-x^2 + 3x + 4) / (x^2 - 2x - 3) = -1
So, the remainder is -1.