Basic Math Examples

$2 q^{4} - 7 q^{3} - 78 q^{2} + 43 q + 40$

Step 1

Factor $2 q^{4} - 7 q^{3} - 78 q^{2} + 43 q + 40$ 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.

$p = \pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm 0.5, \pm 40, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5, \pm 2.5, \pm 8$

Step 1.3

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

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Step 1.3.1

Substitute $- 0.5$ into the polynomial.

$2 {(- 0.5)}^{4} - 7 {(- 0.5)}^{3} - 78 {(- 0.5)}^{2} + 43 \cdot - 0.5 + 40$

Step 1.3.2

Raise $- 0.5$ to the power of $4$ .

$2 \cdot 0.0625 - 7 {(- 0.5)}^{3} - 78 {(- 0.5)}^{2} + 43 \cdot - 0.5 + 40$

Step 1.3.3

Multiply $2$ by $0.0625$ .

$0.125 - 7 {(- 0.5)}^{3} - 78 {(- 0.5)}^{2} + 43 \cdot - 0.5 + 40$

Step 1.3.4

Raise $- 0.5$ to the power of $3$ .

$0.125 - 7 \cdot - 0.125 - 78 {(- 0.5)}^{2} + 43 \cdot - 0.5 + 40$

Step 1.3.5

Multiply $- 7$ by $- 0.125$ .

$0.125 + 0.875 - 78 {(- 0.5)}^{2} + 43 \cdot - 0.5 + 40$

Step 1.3.6

Add $0.125$ and $0.875$ .

$1 - 78 {(- 0.5)}^{2} + 43 \cdot - 0.5 + 40$

Step 1.3.7

Raise $- 0.5$ to the power of $2$ .

$1 - 78 \cdot 0.25 + 43 \cdot - 0.5 + 40$

Step 1.3.8

Multiply $- 78$ by $0.25$ .

$1 - 19.5 + 43 \cdot - 0.5 + 40$

Step 1.3.9

Subtract $19.5$ from $1$ .

$- 18.5 + 43 \cdot - 0.5 + 40$

Step 1.3.10

Multiply $43$ by $- 0.5$ .

$- 18.5 - 21.5 + 40$

Step 1.3.11

Subtract $21.5$ from $- 18.5$ .

$- 40 + 40$

Step 1.3.12

Add $- 40$ and $40$ .

$0$

Step 1.4

Since $- 0.5$ is a known root, divide the polynomial by $2 q + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 q^{4} - 7 q^{3} - 78 q^{2} + 43 q + 40}{2 q + 1}$

Step 1.5

Divide $2 q^{4} - 7 q^{3} - 78 q^{2} + 43 q + 40$ by $2 q + 1$ .

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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 $0$ .

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

Step 1.5.2

Divide the highest order term in the dividend $2 q^{4}$ by the highest order term in divisor $2 q$ .

$q^{3}$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

Step 1.5.3

Multiply the new quotient term by the divisor.

$q^{3}$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

Step 1.5.4

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

$q^{3}$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

Step 1.5.5

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

$q^{3}$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

Step 1.5.6

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

$q^{3}$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

Step 1.5.7

Divide the highest order term in the dividend $- 8 q^{3}$ by the highest order term in divisor $2 q$ .

$q^{3}$

$4 q^{2}$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$q^{3}$	-	$4 q^{2}$
$2 q$	+	$1$		$2 q^{4}$	-	$7 q^{3}$	-	$78 q^{2}$	+	$43 q$	+	$40$
			-	$2 q^{4}$	-	$q^{3}$
					-	$8 q^{3}$	-	$78 q^{2}$
					-	$8 q^{3}$	-	$4 q^{2}$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 q^{3} - 4 q^{2}$

$q^{3}$

$4 q^{2}$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

Step 1.5.10

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

$q^{3}$

$4 q^{2}$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

Step 1.5.11

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

$q^{3}$

$4 q^{2}$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

Step 1.5.12

Divide the highest order term in the dividend $- 74 q^{2}$ by the highest order term in divisor $2 q$ .

$q^{3}$

$4 q^{2}$

$37 q$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

Step 1.5.13

Multiply the new quotient term by the divisor.

$q^{3}$

$4 q^{2}$

$37 q$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

$74 q^{2}$

$37 q$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 74 q^{2} - 37 q$

$q^{3}$

$4 q^{2}$

$37 q$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

$74 q^{2}$

$37 q$

Step 1.5.15

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

$q^{3}$

$4 q^{2}$

$37 q$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

$74 q^{2}$

$37 q$

$80 q$

Step 1.5.16

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

$q^{3}$

$4 q^{2}$

$37 q$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

$74 q^{2}$

$37 q$

$80 q$

$40$

Step 1.5.17

Divide the highest order term in the dividend $80 q$ by the highest order term in divisor $2 q$ .

$q^{3}$

$4 q^{2}$

$37 q$

$40$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

$74 q^{2}$

$37 q$

$80 q$

$40$

Step 1.5.18

Multiply the new quotient term by the divisor.

$q^{3}$

$4 q^{2}$

$37 q$

$40$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

$74 q^{2}$

$37 q$

$80 q$

$40$

$80 q$

$40$

Step 1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $80 q + 40$

$q^{3}$

$4 q^{2}$

$37 q$

$40$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

$74 q^{2}$

$37 q$

$80 q$

$40$

$80 q$

$40$

Step 1.5.20

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

$q^{3}$

$4 q^{2}$

$37 q$

$40$

$2 q$

$1$

$2 q^{4}$

$7 q^{3}$

$78 q^{2}$

$43 q$

$40$

$2 q^{4}$

$q^{3}$

$8 q^{3}$

$78 q^{2}$

$8 q^{3}$

$4 q^{2}$

$74 q^{2}$

$43 q$

$74 q^{2}$

$37 q$

$80 q$

$40$

$80 q$

$40$

$0$

Step 1.5.21

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

$q^{3} - 4 q^{2} - 37 q + 40$

Step 1.6

Write $2 q^{4} - 7 q^{3} - 78 q^{2} + 43 q + 40$ as a set of factors.

$(2 q + 1) (q^{3} - 4 q^{2} - 37 q + 40)$

Step 2

Factor $q^{3} - 4 q^{2} - 37 q + 40$ 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.

$p = \pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

$q = \pm 1$

Step 2.2

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

$\pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

Step 2.3

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

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Step 2.3.1

Substitute $1$ into the polynomial.

$1^{3} - 4 \cdot 1^{2} - 37 \cdot 1 + 40$

Step 2.3.2

Raise $1$ to the power of $3$ .

$1 - 4 \cdot 1^{2} - 37 \cdot 1 + 40$

Step 2.3.3

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

$1 - 4 \cdot 1 - 37 \cdot 1 + 40$

Step 2.3.4

Multiply $- 4$ by $1$ .

$1 - 4 - 37 \cdot 1 + 40$

Step 2.3.5

Subtract $4$ from $1$ .

$- 3 - 37 \cdot 1 + 40$

Step 2.3.6

Multiply $- 37$ by $1$ .

$- 3 - 37 + 40$

Step 2.3.7

Subtract $37$ from $- 3$ .

$- 40 + 40$

Step 2.3.8

Add $- 40$ and $40$ .

$0$

Step 2.4

Since $1$ is a known root, divide the polynomial by $q - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{q^{3} - 4 q^{2} - 37 q + 40}{q - 1}$

Step 2.5

Divide $q^{3} - 4 q^{2} - 37 q + 40$ by $q - 1$ .

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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 $0$ .

$q$

$1$

$q^{3}$

$4 q^{2}$

$37 q$

$40$

Step 2.5.2

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

					$q^{2}$
	$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$

Step 2.5.3

Multiply the new quotient term by the divisor.

				$q^{2}$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			+	$q^{3}$	-	$q^{2}$

Step 2.5.4

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

				$q^{2}$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$

Step 2.5.5

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

				$q^{2}$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$

Step 2.5.6

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

				$q^{2}$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$

Step 2.5.7

Divide the highest order term in the dividend $- 3 q^{2}$ by the highest order term in divisor $q$ .

				$q^{2}$	-	$3 q$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$

Step 2.5.8

Multiply the new quotient term by the divisor.

				$q^{2}$	-	$3 q$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$
					-	$3 q^{2}$	+	$3 q$

Step 2.5.9

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

				$q^{2}$	-	$3 q$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$
					+	$3 q^{2}$	-	$3 q$

Step 2.5.10

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

				$q^{2}$	-	$3 q$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$
					+	$3 q^{2}$	-	$3 q$
							-	$40 q$

Step 2.5.11

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

				$q^{2}$	-	$3 q$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$
					+	$3 q^{2}$	-	$3 q$
							-	$40 q$	+	$40$

Step 2.5.12

Divide the highest order term in the dividend $- 40 q$ by the highest order term in divisor $q$ .

				$q^{2}$	-	$3 q$	-	$40$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$
					+	$3 q^{2}$	-	$3 q$
							-	$40 q$	+	$40$

Step 2.5.13

Multiply the new quotient term by the divisor.

				$q^{2}$	-	$3 q$	-	$40$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$
					+	$3 q^{2}$	-	$3 q$
							-	$40 q$	+	$40$
							-	$40 q$	+	$40$

Step 2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 40 q + 40$

				$q^{2}$	-	$3 q$	-	$40$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$
					+	$3 q^{2}$	-	$3 q$
							-	$40 q$	+	$40$
							+	$40 q$	-	$40$

Step 2.5.15

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

				$q^{2}$	-	$3 q$	-	$40$
$q$	-	$1$		$q^{3}$	-	$4 q^{2}$	-	$37 q$	+	$40$
			-	$q^{3}$	+	$q^{2}$
					-	$3 q^{2}$	-	$37 q$
					+	$3 q^{2}$	-	$3 q$
							-	$40 q$	+	$40$
							+	$40 q$	-	$40$
										$0$

Step 2.5.16

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

$q^{2} - 3 q - 40$

Step 2.6

Write $q^{3} - 4 q^{2} - 37 q + 40$ as a set of factors.

$(2 q + 1) ((q - 1) (q^{2} - 3 q - 40))$

Step 3

Factor $q^{2} - 3 q - 40$ using the AC method.

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Step 3.1

Factor $q^{2} - 3 q - 40$ using the AC method.

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Step 3.1.1

Factor $q^{2} - 3 q - 40$ using the AC method.

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Step 3.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 40$ and whose sum is $- 3$ .

$- 8, 5$

Step 3.1.1.2

Write the factored form using these integers.

$(2 q + 1) ((q - 1) ((q - 8) (q + 5)))$

Step 3.1.2

Remove unnecessary parentheses.

$(2 q + 1) ((q - 1) (q - 8) (q + 5))$

Step 3.2

Remove unnecessary parentheses.

$(2 q + 1) (q - 1) (q - 8) (q + 5)$