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A Remainder Theorem is an approach of Euclidean division of polynomials. Learn about the theorem's proof, Euler's remainder theorem along with solved examples at BYJU'S.
The remainder theorem says "when a polynomial p (x) is divided by a linear polynomial whose zero is x = k, the remainder is given by p (k)". The basic formula to check the division is: Dividend = (Divisor × Quotient) + Remainder. The remainder theorem does not work when the divisor is not linear.
- Polynomials
- The Remainder Theorem
- The Factor Theorem
- Why Is This Useful?
Well, we can also divide polynomials. f(x) ÷ d(x) = q(x) with a remainder of r(x) But it is better to write it as a sum like this: Like in this example using Polynomial Long Division(the method we want to avoid): And there is a key feature: Say we divide by a polynomial of degree 1 (such as "x−3") the remainder will have degree 0(in other words a c...
When we divide f(x) by the simple polynomial x−cwe get: f(x) = (x−c) q(x) + r(x) x−c is degree 1, so r(x) must have degree 0, so it is just some constant r: f(x) = (x−c) q(x) + r Now see what happens when we have x equal to c: So we get this: So to find the remainder after dividing by x-cwe don't need to do any division: Just calculate f(c) Let us ...
Now ... We see this when dividing whole numbers. For example 60 ÷ 20 = 3 with no remainder. So 20 must be a factor of 60. And so we have:
Knowing that x−c is a factor is the same as knowing that c is a root (and vice versa). For one thing, it means that we can quickly check if (x−c) is a factor of the polynomial.
The Polynomial Remainder Theorem simplifies the process of finding the remainder when dividing a polynomial by \[x - a\]. Instead of long division, you just evaluate the polynomial at \[a\]. This method saves time and space, making polynomial division more manageable.
- 4 min
- He was just plugging in _a_ (in this case, 2) for _x_. The first term of the equation is -3_x_^3. When _x_ is plugged in, it is -3*2^3, which is al...
- If you have (x+a), then the "a" value will actually be "-a" because (x+(-a))=(x-a).
- If the remainder is 0, then you know that the divisior is a factor of the dividend (they are divisble).
- With polynomial division, you can get remainders that are negative. In numeric long division, you would not have a negative remainder.
- I think that it could work. Another way to find the remainder is to set the x - a to term equal to 0 and then solve for x. After this, you just plu...
- It depends on the context of the problem (if it asks for remainder _term_); but usually, and in this problem, it should be expressed as -27, accord...
- Function of x is basically a processing machine. You give it an input, and it gives out one output and one only. For example if f(x) = x+1, you can...
- There's not much of a difference. The existence of a remainder essentially implies that if you added that value to the original equation (irrespect...
- Think about when you're dividing normal numbers that don't go into each other easily, like 9 / 4. In this problem, the remainder would only be 1. H...
The polynomial remainder theorem says that for a polynomial p(x) and a number a, the remainder on division by (x-a) is p(a). This might not be very clear right now, but you will understand this much better after watching these examples.
- 6 min
- According to the polynomial remainder theorem, when you divide the polynomial function, P(x), by x-a, then the remainder will be P(a). In this case...
- If you divide a polynomial by (x-a) the remainder to that is the same as solving for f(a), where f is the polynomial. So in other words you plug a...
- When you divide by a polynomial of degree n, you get a remainder of degree n-1 or less. So here, your remainder would be of the form ax+b. If you'r...
- A polynomial 𝑝(𝑥) divided by (𝑎𝑥 − 𝑏) will have a remainder of 𝑝(𝑏∕𝑎).
- P(x) = ((divisor)*(quotient)) + (remainder) P(x) = (x-a)*(q) + r Now, if we take x = a, P(a) = (a-a)*q + r P(a) = 0 + r = r If you had taken some o...
- It's really not that hard once you get it. Just find the x value of the divisor (To find this just find the value of "x" which will make the diviso...
27 de may. de 2024 · The Remainder Theorem states that if a polynomial f(x) of degree n (≥ 1) is divided by a linear polynomial (a polynomial of degree 1) g(x) of the form (x – a), the remainder of this division is the same as the value obtained by substituting r(x) = f(a) into the polynomial f(x).
Learn how to determine if an expression is a factor of a polynomial by dividing the polynomial by the expression. If the remainder is zero, the expression is a factor. The video also demonstrates how to quickly calculate the remainder using the theorem.
- 3 min
