Determine if the Given Expression is a Factor

$x^{3} - 3 x^{2} - 2 x + 6$ , $x - 4$

Step 1

If $x - 4$ is a factor of the polynomial, then it must be a root of the polynomial. To determine if the factor is a root, first find the value of $x$ when the factor $x - 4$ is equal to $0$ .

$x - 4 = 0$

Step 2

Solve the equation for $x$ .

$x = 4$

Step 3

Replace $x$ with $4$ in $x^{3} - 3 x^{2} - 2 x + 6$ to find if it is a root of the polynomial.

${(4)}^{3} - 3 {(4)}^{2} - 2 \cdot 4 + 6$

Step 4

Simplify each term.

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Step 4.1

Raise $4$ to the power of $3$ .

$64 - 3 {(4)}^{2} - 2 \cdot 4 + 6$

Step 4.2

Raise $4$ to the power of $2$ .

$64 - 3 \cdot 16 - 2 \cdot 4 + 6$

Step 4.3

Multiply $- 3$ by $16$ .

$64 - 48 - 2 \cdot 4 + 6$

Step 4.4

Multiply $- 2$ by $4$ .

$64 - 48 - 8 + 6$

Step 5

Simplify by adding and subtracting.

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Step 5.1

Subtract $48$ from $64$ .

$16 - 8 + 6$

Step 5.2

Subtract $8$ from $16$ .

$8 + 6$

Step 5.3

Add $8$ and $6$ .

$14$

Step 6

Since $x = 4$ is not a root of the polynomial, $x - 4$ is not a factor of the polynomial.

$x - 4$ is not a factor of $x^{3} - 3 x^{2} - 2 x + 6$

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Determine if the Given Expression is a Factor

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