Algebra Examples

$[\begin{array}{c} x & - 1 & - 2 \\ x & 4 & x \\ 7 & 0 & x^{2} \end{array}]$

Step 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in column $1$ by its cofactor and add.

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Step 1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 4 & x \\ 0 & x^{2} \end{array} |$

Step 1.4

Multiply element $a_{11}$ by its cofactor.

$x | \begin{array}{c} 4 & x \\ 0 & x^{2} \end{array} |$

Step 1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} - 1 & - 2 \\ 0 & x^{2} \end{array} |$

Step 1.6

Multiply element $a_{21}$ by its cofactor.

$- x | \begin{array}{c} - 1 & - 2 \\ 0 & x^{2} \end{array} |$

Step 1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$

Step 1.8

Multiply element $a_{31}$ by its cofactor.

$7 | \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$

Step 1.9

Add the terms together.

$x | \begin{array}{c} 4 & x \\ 0 & x^{2} \end{array} | - x | \begin{array}{c} - 1 & - 2 \\ 0 & x^{2} \end{array} | + 7 | \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 4 & x \\ 0 & x^{2} \end{array} |$ .

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Step 2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$x (4 x^{2} + 0 x) - x | \begin{array}{c} - 1 & - 2 \\ 0 & x^{2} \end{array} | + 7 | \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Multiply $0$ by $x$ .

$x (4 x^{2} + 0) - x | \begin{array}{c} - 1 & - 2 \\ 0 & x^{2} \end{array} | + 7 | \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$

Step 2.2.2

Add $4 x^{2}$ and $0$ .

$x (4 x^{2}) - x | \begin{array}{c} - 1 & - 2 \\ 0 & x^{2} \end{array} | + 7 | \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$

Step 3

Evaluate $| \begin{array}{c} - 1 & - 2 \\ 0 & x^{2} \end{array} |$ .

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Step 3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$x (4 x^{2}) - x (- x^{2} + 0 \cdot - 2) + 7 | \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$

Step 3.2

Simplify the determinant.

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Step 3.2.1

Multiply $0$ by $- 2$ .

$x (4 x^{2}) - x (- x^{2} + 0) + 7 | \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$

Step 3.2.2

Add $- x^{2}$ and $0$ .

$x (4 x^{2}) - x (- x^{2}) + 7 | \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$

Step 4

Evaluate $| \begin{array}{c} - 1 & - 2 \\ 4 & x \end{array} |$ .

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Step 4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$x (4 x^{2}) - x (- x^{2}) + 7 (- x - 4 \cdot - 2)$

Step 4.2

Multiply $- 4$ by $- 2$ .

$x (4 x^{2}) - x (- x^{2}) + 7 (- x + 8)$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

$4 x \cdot x^{2} - x (- x^{2}) + 7 (- x + 8)$

Step 5.1.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.1.2.1

Move $x^{2}$ .

$4 (x^{2} x) - x (- x^{2}) + 7 (- x + 8)$

Step 5.1.2.2

Multiply $x^{2}$ by $x$ .

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Step 5.1.2.2.1

Raise $x$ to the power of $1$ .

$4 (x^{2} x^{1}) - x (- x^{2}) + 7 (- x + 8)$

Step 5.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x^{2 + 1} - x (- x^{2}) + 7 (- x + 8)$

Step 5.1.2.3

Add $2$ and $1$ .

$4 x^{3} - x (- x^{2}) + 7 (- x + 8)$

Step 5.1.3

Rewrite using the commutative property of multiplication.

$4 x^{3} - 1 \cdot - 1 x \cdot x^{2} + 7 (- x + 8)$

Step 5.1.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.1.4.1

Move $x^{2}$ .

$4 x^{3} - 1 \cdot - 1 (x^{2} x) + 7 (- x + 8)$

Step 5.1.4.2

Multiply $x^{2}$ by $x$ .

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Step 5.1.4.2.1

Raise $x$ to the power of $1$ .

$4 x^{3} - 1 \cdot - 1 (x^{2} x^{1}) + 7 (- x + 8)$

Step 5.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x^{3} - 1 \cdot - 1 x^{2 + 1} + 7 (- x + 8)$

Step 5.1.4.3

Add $2$ and $1$ .

$4 x^{3} - 1 \cdot - 1 x^{3} + 7 (- x + 8)$

Step 5.1.5

Multiply $- 1$ by $- 1$ .

$4 x^{3} + 1 x^{3} + 7 (- x + 8)$

Step 5.1.6

Multiply $x^{3}$ by $1$ .

$4 x^{3} + x^{3} + 7 (- x + 8)$

Step 5.1.7

Apply the distributive property.

$4 x^{3} + x^{3} + 7 (- x) + 7 \cdot 8$

Step 5.1.8

Multiply $- 1$ by $7$ .

$4 x^{3} + x^{3} - 7 x + 7 \cdot 8$

Step 5.1.9

Multiply $7$ by $8$ .

$4 x^{3} + x^{3} - 7 x + 56$

Step 5.2

Add $4 x^{3}$ and $x^{3}$ .

$5 x^{3} - 7 x + 56$