Polynomial f(x) has a remainder of 3 when divided by x-2 and a remainder of -5 when it is divided by x+2. Determine the remainder when the polynomial is divided by x^2-4.
The remainder when f is divided by (x+2)(x−2) is at most a first degree polynomial. That means that f(x)= g(x)⋅(x+2)(x−2) + (ax+b) for some real numbers a, and b
f(x) = g(x) (x + 2)(x - 2) + ax + b.
We are told that when f(x) is divided by x-2 the remainder is 3 , so
f(2) = g(2) (4)(0) + (2a) + b
3 = 0 + 2a + b
2a + b = 3 **
when f(x) is divided by x+2 the remainder is -5
f(-2) = g(-2) (0)(-4) - 2a + b
-5 = 0 - 2a + b
2a - b = 5 ***
add ** and ***
4a = 8
a = 2 , then b = -1
so the remainder is 2x - 1
Well, it seems this polynomial really knows how to spread the love (or the remainders, in this case). Let's see if we can keep up with its shenanigans!
We know that when the polynomial is divided by x-2, it leaves a remainder of 3. So, we can say that f(2) = 3. Similarly, when it's divided by x+2, it leaves a remainder of -5. This means that f(-2) = -5.
Now, we want to find the remainder when the polynomial is divided by x^2-4. Well, let's take a look at x^2-4 for a moment. Notice anything, uh, square about it? That's right, it's a difference of squares, (x+2)(x-2).
So, we have the remainders when f(x) is divided by both x-2 and x+2. And we also have the factorization of x^2-4 as (x+2)(x-2). It's starting to feel like a family reunion here!
Now, remembering our good old friend, the Remainder Theorem, we can use it to find the remainder. We'll substitute x = 2 and x = -2 into the polynomial, and that will give us the remainders.
Let's do the math before anyone says "clowning around":
f(2) = 3
f(-2) = -5
Now, let's find the difference between these remainders:
3 - (-5) = 8
So, the remainder when the polynomial is divided by x^2-4 is 8. Ta-da! We made it through the circus of polynomial remainders! I hope this answer has brought a smile to your face.
To find the remainder when the polynomial is divided by x^2-4, we need to determine the values of the coefficients in the quotient when f(x) is divided by x^2-4.
Given that f(x) has a remainder of 3 when divided by x-2, we can write the equation as:
f(x) = (x-2)q(x) + 3 ---- (1)
Similarly, f(x) has a remainder of -5 when divided by x+2, so we can write the equation as:
f(x) = (x+2)p(x) - 5 ---- (2)
Now, let's find the quotient when f(x) is divided by x^2-4. We can express x^2-4 as a product of (x+2) and (x-2):
x^2 - 4 = (x+2)(x-2)
Substituting this into equation (2) and rearranging, we get:
f(x) = (x+2)p(x) - 5
f(x) = ((x+2)(x-2))p(x) - 5
f(x) = (x^2 - 4)p(x) - 5
f(x) = x^2p(x) - 4p(x) - 5 ---- (3)
We want to find the remainder when f(x) is divided by x^2-4, so we need to express f(x) in terms of (x^2 - 4) and a remainder term. Let's express f(x) as:
f(x) = (x^2 - 4)q'(x) + r ---- (4)
where q'(x) is the new quotient and r is the remainder.
Comparing equations (3) and (4), we see that the remainder r is given by:
r = -4p(x) - 5
Therefore, the remainder when the polynomial is divided by x^2-4 is -4p(x) - 5.
To determine the remainder when the polynomial is divided by x^2-4, we need to use the Remainder Theorem.
The Remainder Theorem states that when a polynomial f(x) is divided by x-a, the remainder is equal to f(a).
In this case, we are given that the polynomial f(x) has a remainder of 3 when divided by x-2 and a remainder of -5 when divided by x+2.
Let's first find the value of f(2):
f(2) represents the remainder when f(x) is divided by x-2.
Since we are given that the remainder is 3, we have f(2) = 3.
Next, let's find the value of f(-2):
f(-2) represents the remainder when f(x) is divided by x+2.
Since we are given that the remainder is -5, we have f(-2) = -5.
Now, we can use these values to find the remainder when f(x) is divided by x^2-4.
Since x^2-4 can be factored as (x-2)(x+2), we can use the Remainder Theorem for both factors.
To find the remainder when f(x) is divided by x-2, we use f(2) = 3.
To find the remainder when f(x) is divided by x+2, we use f(-2) = -5.
So, f(x) = (x-2)q(x) + 3 and f(x) = (x+2)r(x) - 5, where q(x) and r(x) are quotient polynomials.
Now, we can equate these two expressions for f(x) to find the remainder when f(x) is divided by x^2-4:
(x-2)q(x) + 3 = (x+2)r(x) - 5
We can simplify this equation to get:
(x-2)q(x) - (x+2)r(x) = -8
We can see that the remainder when f(x) is divided by x^2-4 is -8.
So, the remainder when f(x) is divided by x^2-4 is -8.