Solving Polynomial Remainder Problem
This solution explains how to find the remainder when the polynomial $P(x) = x^4 + x^3 - x^2 + x + 1$ is divided by the binomial $x - 3$. We will use the Remainder Theorem.
Remainder Theorem Application
The Remainder Theorem states that if a polynomial $P(x)$ is divided by a linear divisor of the form $x - c$, the remainder is equal to the value of the polynomial evaluated at $x = c$, which is $P(c)$.
Step-by-Step Calculation
In this problem, the polynomial is $P(x) = x^4 + x^3 - x^2 + x + 1$ and the divisor is $x - 3$. Here, $c = 3$.
- Identify the value of $c$ from the divisor $x - c$. In this case, $c = 3$.
- Substitute $x = 3$ into the polynomial $P(x)$:
$P(3) = (3)^4 + (3)^3 - (3)^2 + (3) + 1$
- Calculate the powers of 3:
$P(3) = 81 + 27 - 9 + 3 + 1$
- Perform the addition and subtraction:
$P(3) = 108 - 9 + 4$
$P(3) = 99 + 4$
$P(3) = 103$
Therefore, the remainder when $x^4 + x^3 - x^2 + x + 1$ is divided by $x - 3$ is 103.
Similar Questions
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Important Questions from Polynomials
If (x + y) 3+ 8 (x - y) 3= (3x + Ay) (3x 2+ Bxy + Cy 2), then the value of A + B + C is:
Given that x 8- 34x 4+ 1 = 0, x > 0. What is the value of (x 3+ x -3 )?
If \(x - \frac 3 x = 6,\; x \ne 0,\) then the value of \(\frac {x^4 - \frac {27}{x^2}}{x^2 - 3x - 3}\) is:
If \(x\left(3 - \frac 2 x\right) = \frac 3 x,\) then the value of \(x^3 - \frac 1 {x^3}\) is equal to:
The coefficient of x in (x – 3y) 3is:
