Algebra Examples

$[\begin{array}{c} x & 0 & - 5 \\ 3 & x & x^{2} \\ - 4 & x & 1 \end{array}]$

Set up the determinant by breaking it into smaller components.

$0 | \begin{array}{c} 3 & x^{2} \\ - 4 & 1 \end{array} | + x | \begin{array}{c} x & - 5 \\ - 4 & 1 \end{array} | - x | \begin{array}{c} x & - 5 \\ 3 & x^{2} \end{array} |$

Since the matrix is multiplied by $0$ , the determinant is $0$ .

$0 + x | \begin{array}{c} x & - 5 \\ - 4 & 1 \end{array} | - x | \begin{array}{c} x & - 5 \\ 3 & x^{2} \end{array} |$

Find the determinant of $x | \begin{array}{c} x & - 5 \\ - 4 & 1 \end{array} |$ .

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

$0 + x ((x) (1) + 4 \cdot - 5) - x | \begin{array}{c} x & - 5 \\ 3 & x^{2} \end{array} |$

Simplify the determinant.

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Simplify each term.

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Multiply $x$ by $1$ .

$0 + x (x + 4 \cdot - 5) - x | \begin{array}{c} x & - 5 \\ 3 & x^{2} \end{array} |$

Multiply $4$ by $- 5$ .

$0 + x (x - 20) - x | \begin{array}{c} x & - 5 \\ 3 & x^{2} \end{array} |$

Simplify by multiplying through.

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Apply the distributive property.

$0 + x \cdot x + x \cdot - 20 - x | \begin{array}{c} x & - 5 \\ 3 & x^{2} \end{array} |$

Simplify the expression.

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Multiply $x$ by $x$ .

$0 + x^{2} + x \cdot - 20 - x | \begin{array}{c} x & - 5 \\ 3 & x^{2} \end{array} |$

Move $- 20$ to the left of $x$ .

$0 + x^{2} - 20 x - x | \begin{array}{c} x & - 5 \\ 3 & x^{2} \end{array} |$

Find the determinant of $- x | \begin{array}{c} x & - 5 \\ 3 & x^{2} \end{array} |$ .

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

$0 + x^{2} - 20 x - x ((x) (x^{2}) - 3 \cdot - 5)$

Simplify the determinant.

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Simplify each term.

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

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

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

$0 + x^{2} - 20 x - x (x \cdot x^{2} - 3 \cdot - 5)$

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

$0 + x^{2} - 20 x - x (x^{1 + 2} - 3 \cdot - 5)$

Add $1$ and $2$ .

$0 + x^{2} - 20 x - x (x^{3} - 3 \cdot - 5)$

Multiply $- 3$ by $- 5$ .

$0 + x^{2} - 20 x - x (x^{3} + 15)$

Apply the distributive property.

$0 + x^{2} - 20 x - x \cdot x^{3} - x \cdot 15$

Multiply $x$ by $x^{3}$ by adding the exponents.

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Move $x^{3}$ .

$0 + x^{2} - 20 x - (x^{3} x) - x \cdot 15$

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

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

$0 + x^{2} - 20 x - (x^{3} x) - x \cdot 15$

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

$0 + x^{2} - 20 x - x^{3 + 1} - x \cdot 15$

Add $3$ and $1$ .

$0 + x^{2} - 20 x - x^{4} - x \cdot 15$

Multiply $15$ by $- 1$ .

$0 + x^{2} - 20 x - x^{4} - 15 x$

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

$x^{2} - 20 x - x^{4} - 15 x$

Subtract $15 x$ from $- 20 x$ .

$x^{2} - x^{4} - 35 x$