Question34. $x^{3}-6 x^{2}+11 x-6 ; x-3$

Studdy Solution

STEP 1

1. We are given a polynomial $x^3 - 6x^2 + 11x - 6$ and a divisor $x - 3$ .
2. We need to perform polynomial division to determine if $x - 3$ is a factor of the polynomial.
3. If the remainder of the division is zero, then $x - 3$ is a factor.

STEP 2

1. Set up the polynomial division.
2. Perform the polynomial division.
3. Determine if there is a remainder.
4. Conclude whether $x - 3$ is a factor.

STEP 3

Set up the polynomial division of $x^3 - 6x^2 + 11x - 6$ by $x - 3$ .
We will use synthetic division since the divisor is linear and of the form $x - c$ .

STEP 4

Perform synthetic division:
1. Write down the coefficients of the polynomial: $1, -6, 11, -6$ .
2. Use the root $3$ from $x - 3$ for synthetic division.
$\begin{array}{r|rrrr} 3 & 1 & -6 & 11 & -6 \\ & & 3 & -9 & 6 \\ \hline & 1 & -3 & 2 & 0 \\ \end{array}$

STEP 5

Determine if there is a remainder:
The last number in the bottom row is the remainder. In this case, the remainder is $0$ .

STEP 6

Conclude whether $x - 3$ is a factor:
Since the remainder is $0$ , $x - 3$ is indeed a factor of the polynomial $x^3 - 6x^2 + 11x - 6$ .
The polynomial $x^3 - 6x^2 + 11x - 6$ is divisible by $x - 3$ with no remainder, confirming that $x - 3$ is a factor.

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