Provide the numerator and denominator polynomial and the calculator will determine their remainder by using the remainder theorem.
The rest theorem calculator is loose on line tool that helps you to calculate the remainder of given polynomial expressions through the rest theorem.
Our aspect theorem calculator offers little by little calculations of the aspect of division. right here you can apprehend a way to locate the the rest of a polynomial the use of the formulation.
In algebra, the remainder theorem or little Bezout’s theorem is an utility of Euclidean division of different expressions, that is located by way of Etienne Bezout. It states when an expression is split via a element x-j, then the the rest of the division is same to f(j).
when the polynomial f(x) is divisible by using a linear issue of the form x-j, the theorem will be utilized by the remainder theorem calculator. if you want to do those calculations by hand, then comply with the instructions beneath and use them to resolve the rest of the polynomial expression in a few minutes.
Example
Solve \( (x^4 + 8x^3 - 6x^2 + 5x - 11) \) with denominator \( (x + 2) \) using the remainder theorem.
Solution: Given values are
$$f(x) = x^4 + 8x^3 - 6x^2 + 5x - 11$$
Since \( x + 2 \) is in the form of \( x - (-2) \),
Then \( c = -2 \)
$$f(-2) = (-2)^4 + 8(-2)^3 - 6(-2)^2 + 5(-2) - 11$$
$$= 16 + 8(-8) - 6(4) + 5(-2) - 11$$
$$= 16 - 64 - 24 - 10 - 11$$
$$= -93$$
The remainder of the given expression is \( -93 \).
The the rest calculator calculates:
| Property | Formula | Example Calculation |
|---|---|---|
| Remainder Theorem | If a polynomial f(x) is divided by (x - c), the remainder is f(c). | For f(x) = x² - 5x + 6, remainder when divided by (x - 2) is f(2) = 2² - 5(2) + 6 = 4 - 10 + 6 = 0. |
| Divide Polynomial by Linear Binomial | Remainder = f(c) | If f(x) = x³ - 4x² + 3x - 2 and the divisor is (x - 1), then f(1) = 1³ - 4(1)² + 3(1) - 2 = 1 - 4 + 3 - 2 = -2. |
| Dividing Polynomial by (x - c) | The remainder is the value of the polynomial at x = c. | For f(x) = 2x³ + 3x² - x - 5 and divisor (x + 1), calculate f(-1) = 2(-1)³ + 3(-1)² - (-1) - 5 = -2 + 3 + 1 - 5 = -3. |
| Finding the Remainder with Synthetic Division | Perform synthetic division with c as the value. | If f(x) = x² + 3x - 4 and dividing by (x - 1), synthetic division gives remainder = f(1) = 1² + 3(1) - 4 = 0. |
| Factor Theorem | If f(c) = 0, then (x - c) is a factor of f(x). | For f(x) = x² - 5x + 6, since f(2) = 0, (x - 2) is a factor of f(x). |
| Dividing Polynomial by Non-Linear Polynomial | Remainder is obtained by applying the Remainder Theorem at appropriate values. | If f(x) = x³ - x² + 2x - 3 and divisor (x² - 1), we find the remainder by evaluating at appropriate x values (use direct substitution of values in polynomial). |
| Remainder of Polynomial Divided by (x - c) | Remainder is the value of the polynomial at x = c. | If f(x) = 2x⁴ - 3x³ + x - 4 and divisor (x - 1), then f(1) = 2(1)⁴ - 3(1)³ + 1 - 4 = 2 - 3 + 1 - 4 = -4. |
| Evaluating Polynomial at Specific Value | Substitute the value of x directly into f(x). | For f(x) = x² + 2x + 1, remainder when divided by (x - 1) is f(1) = 1² + 2(1) + 1 = 4. |
| Remainder for Polynomial Division | Find the remainder by substituting x = c into the polynomial. | For f(x) = x⁴ - 4x³ + 3x² - 2x + 1 and divisor (x - 3), f(3) = 3⁴ - 4(3³) + 3(3²) - 2(3) + 1 = 81 - 108 + 27 - 6 + 1 = -5. |
| Using Remainder Theorem for Evaluation | Substitute c into f(x) to evaluate the remainder. | If f(x) = x³ - 6x² + 11x - 6 and divisor (x - 2), f(2) = 2³ - 6(2²) + 11(2) - 6 = 8 - 24 + 22 - 6 = 0. |
Remainder Theorem postulates that when dividing a polynomial by a first-degree divider, the residual of said division equals the value of the polynomial at the designated c.
To apply the Remainder Theorem, replace the value c into the polynomial. The result is the remaining when the polynomial is divided by x-c.
Yes, the Remainder Theorem applies to any polynomial when divided by a linear divider in the shape x-c.
To use the calculator, type in the equation and the number for c, and the calculator will count the polynomial at c and provide the leftover.
Absolutely, the Remainder Theorem applies specifically when dividing by a linear divisor shaped like x-c.
If the remainder is nil, then the divider (x-c) is an element of the polynomial.
Both methods can solve the division of polynomials, but the Remainder Theorem is a easier method for determining the remaining.
Affirmative, the Remainder Theorem functions for expressions with complex coefficients as well as integers.
Yes, the Remainder Theorem applies to polynomials of any degree when the division involves a linear divisor.
