Finite Math Examples

$f (x) = x^{4} - 3 x^{3} - 6 x^{2} + 28 x - 24$

Step 1

Factor $x^{4} - 3 x^{3} - 6 x^{2} + 28 x - 24$ using the rational roots test.

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Step 1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$f (x) = p = \pm 1, \pm 24, \pm 2, \pm 12, \pm 3, \pm 8, \pm 4, \pm 6$

$q = \pm 1$

Step 1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$f (x) = \pm 1, \pm 24, \pm 2, \pm 12, \pm 3, \pm 8, \pm 4, \pm 6$

Step 1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 1.3.1

Substitute $2$ into the polynomial.

$f (x) = 2^{4} - 3 \cdot 2^{3} - 6 \cdot 2^{2} + 28 \cdot 2 - 24$

Step 1.3.2

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

$f (x) = 16 - 3 \cdot 2^{3} - 6 \cdot 2^{2} + 28 \cdot 2 - 24$

Step 1.3.3

Raise $2$ to the power of $3$ .

$f (x) = 16 - 3 \cdot 8 - 6 \cdot 2^{2} + 28 \cdot 2 - 24$

Step 1.3.4

Multiply $- 3$ by $8$ .

$f (x) = 16 - 24 - 6 \cdot 2^{2} + 28 \cdot 2 - 24$

Step 1.3.5

Subtract $24$ from $16$ .

$f (x) = - 8 - 6 \cdot 2^{2} + 28 \cdot 2 - 24$

Step 1.3.6

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

$f (x) = - 8 - 6 \cdot 4 + 28 \cdot 2 - 24$

Step 1.3.7

Multiply $- 6$ by $4$ .

$f (x) = - 8 - 24 + 28 \cdot 2 - 24$

Step 1.3.8

Subtract $24$ from $- 8$ .

$f (x) = - 32 + 28 \cdot 2 - 24$

Step 1.3.9

Multiply $28$ by $2$ .

$f (x) = - 32 + 56 - 24$

Step 1.3.10

Add $- 32$ and $56$ .

$f (x) = 24 - 24$

Step 1.3.11

Subtract $24$ from $24$ .

$f (x) = 0$

Step 1.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$f (x) = \frac{x^{4} - 3 x^{3} - 6 x^{2} + 28 x - 24}{x - 2}$

Step 1.5

Divide $x^{4} - 3 x^{3} - 6 x^{2} + 28 x - 24$ by $x - 2$ .

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Step 1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$f (x) =$

Step 1.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$f (x) =$

Step 1.5.3

Multiply the new quotient term by the divisor.

$f (x) =$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 2 x^{3}$

$f (x) =$

Step 1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$f (x) =$

Step 1.5.6

Pull the next terms from the original dividend down into the current dividend.

$f (x) =$

Step 1.5.7

Divide the highest order term in the dividend $- x^{3}$ by the highest order term in divisor $x$ .

$f (x) =$

Step 1.5.8

Multiply the new quotient term by the divisor.

$f (x) =$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{3} + 2 x^{2}$

$f (x) =$

Step 1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$f (x) =$

Step 1.5.11

Pull the next terms from the original dividend down into the current dividend.

$f (x) =$

Step 1.5.12

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

$f (x) =$

Step 1.5.13

Multiply the new quotient term by the divisor.

$f (x) =$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} + 16 x$

$f (x) =$

Step 1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$f (x) =$

Step 1.5.16

Pull the next terms from the original dividend down into the current dividend.

$f (x) =$

Step 1.5.17

Divide the highest order term in the dividend $12 x$ by the highest order term in divisor $x$ .

$f (x) =$

Step 1.5.18

Multiply the new quotient term by the divisor.

$f (x) =$

Step 1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $12 x - 24$

$f (x) =$

Step 1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$f (x) =$

Step 1.5.21

Since the remander is $0$ , the final answer is the quotient.

$f (x) = x^{3} - x^{2} - 8 x + 12$

Step 1.6

Write $x^{4} - 3 x^{3} - 6 x^{2} + 28 x - 24$ as a set of factors.

$f (x) = (x - 2) (x^{3} - x^{2} - 8 x + 12)$

Step 2

Factor $x^{3} - x^{2} - 8 x + 12$ using the rational roots test.

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Step 2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$f (x) = p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1$

Step 2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$f (x) = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Step 2.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 2.3.1

Substitute $2$ into the polynomial.

$f (x) = 2^{3} - 2^{2} - 8 \cdot 2 + 12$

Step 2.3.2

Raise $2$ to the power of $3$ .

$f (x) = 8 - 2^{2} - 8 \cdot 2 + 12$

Step 2.3.3

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

$f (x) = 8 - 1 \cdot 4 - 8 \cdot 2 + 12$

Step 2.3.4

Multiply $- 1$ by $4$ .

$f (x) = 8 - 4 - 8 \cdot 2 + 12$

Step 2.3.5

Subtract $4$ from $8$ .

$f (x) = 4 - 8 \cdot 2 + 12$

Step 2.3.6

Multiply $- 8$ by $2$ .

$f (x) = 4 - 16 + 12$

Step 2.3.7

Subtract $16$ from $4$ .

$f (x) = - 12 + 12$

Step 2.3.8

Add $- 12$ and $12$ .

$f (x) = 0$

Step 2.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$f (x) = \frac{x^{3} - x^{2} - 8 x + 12}{x - 2}$

Step 2.5

Divide $x^{3} - x^{2} - 8 x + 12$ by $x - 2$ .

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Step 2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$f (x) =$

Step 2.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

$f (x) =$

Step 2.5.3

Multiply the new quotient term by the divisor.

$f (x) =$

Step 2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 2 x^{2}$

$f (x) =$

Step 2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$f (x) =$

Step 2.5.6

Pull the next terms from the original dividend down into the current dividend.

$f (x) =$

Step 2.5.7

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

$f (x) =$

Step 2.5.8

Multiply the new quotient term by the divisor.

$f (x) =$

Step 2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} - 2 x$

$f (x) =$

Step 2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$f (x) =$

Step 2.5.11

Pull the next terms from the original dividend down into the current dividend.

$f (x) =$

Step 2.5.12

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

$f (x) =$

Step 2.5.13

Multiply the new quotient term by the divisor.

$f (x) =$

Step 2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x + 12$

$f (x) =$

Step 2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$f (x) =$

Step 2.5.16

Since the remander is $0$ , the final answer is the quotient.

$f (x) = x^{2} + x - 6$

Step 2.6

Write $x^{3} - x^{2} - 8 x + 12$ as a set of factors.

$f (x) = (x - 2) ((x - 2) (x^{2} + x - 6))$

Step 3

Factor $x^{2} + x - 6$ using the AC method.

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Step 3.1

Factor $x^{2} + x - 6$ using the AC method.

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Step 3.1.1

Consider the form $x^{2} + i x + c$ . Find a pair of integers whose product is $c$ and whose sum is $i$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$f (x) = - 2, 3$

Step 3.1.2

Write the factored form using these integers.

$f (x) = (x - 2) ((x - 2) ((x - 2) (x + 3)))$

Step 3.2

Remove unnecessary parentheses.

$f (x) = (x - 2) ((x - 2) (x - 2) (x + 3))$

Step 4

Combine like factors.

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Step 4.1

Combine like factors.

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Step 4.1.1

Raise $x - 2$ to the power of $1$ .

$f (x) = (x - 2) ((x - 2) (x - 2) (x + 3))$

Step 4.1.2

Raise $x - 2$ to the power of $1$ .

$f (x) = (x - 2) ((x - 2) (x - 2) (x + 3))$

Step 4.1.3

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

$f (x) = (x - 2) ({(x - 2)}^{1 + 1} (x + 3))$

Step 4.1.4

Add $1$ and $1$ .

$f (x) = (x - 2) ({(x - 2)}^{2} (x + 3))$

Step 4.2

Remove unnecessary parentheses.

$f (x) = (x - 2) {(x - 2)}^{2} (x + 3)$

Step 5

Combine like factors.

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Step 5.1

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

$f (x) = {(x - 2)}^{1 + 2} (x + 3)$

Step 5.2

Add $1$ and $2$ .

$f (x) = {(x - 2)}^{3} (x + 3)$