Basic Math Examples

$4 t^{3} + 3 t^{2} - 5 t - 2$

Step 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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 4, \pm 2$

Step 2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 2$

Step 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 3.1

Substitute $1$ into the polynomial.

$4 \cdot 1^{3} + 3 \cdot 1^{2} - 5 \cdot 1 - 2$

Step 3.2

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

$4 \cdot 1 + 3 \cdot 1^{2} - 5 \cdot 1 - 2$

Step 3.3

Multiply $4$ by $1$ .

$4 + 3 \cdot 1^{2} - 5 \cdot 1 - 2$

Step 3.4

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

$4 + 3 \cdot 1 - 5 \cdot 1 - 2$

Step 3.5

Multiply $3$ by $1$ .

$4 + 3 - 5 \cdot 1 - 2$

Step 3.6

Add $4$ and $3$ .

$7 - 5 \cdot 1 - 2$

Step 3.7

Multiply $- 5$ by $1$ .

$7 - 5 - 2$

Step 3.8

Subtract $5$ from $7$ .

$2 - 2$

Step 3.9

Subtract $2$ from $2$ .

$0$

Step 4

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

$\frac{4 t^{3} + 3 t^{2} - 5 t - 2}{t - 1}$

Step 5

Divide $4 t^{3} + 3 t^{2} - 5 t - 2$ by $t - 1$ .

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Step 5.1

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

$t$

$1$

$4 t^{3}$

$3 t^{2}$

$5 t$

$2$

Step 5.2

Divide the highest order term in the dividend $4 t^{3}$ by the highest order term in divisor $t$ .

					$4 t^{2}$
	$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$

Step 5.3

Multiply the new quotient term by the divisor.

				$4 t^{2}$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			+	$4 t^{3}$	-	$4 t^{2}$

Step 5.4

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

				$4 t^{2}$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$

Step 5.5

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

				$4 t^{2}$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$

Step 5.6

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

				$4 t^{2}$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$

Step 5.7

Divide the highest order term in the dividend $7 t^{2}$ by the highest order term in divisor $t$ .

				$4 t^{2}$	+	$7 t$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$

Step 5.8

Multiply the new quotient term by the divisor.

				$4 t^{2}$	+	$7 t$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$
					+	$7 t^{2}$	-	$7 t$

Step 5.9

The expression needs to be subtracted from the dividend, so change all the signs in $7 t^{2} - 7 t$

				$4 t^{2}$	+	$7 t$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$
					-	$7 t^{2}$	+	$7 t$

Step 5.10

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

				$4 t^{2}$	+	$7 t$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$
					-	$7 t^{2}$	+	$7 t$
							+	$2 t$

Step 5.11

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

				$4 t^{2}$	+	$7 t$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$
					-	$7 t^{2}$	+	$7 t$
							+	$2 t$	-	$2$

Step 5.12

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

				$4 t^{2}$	+	$7 t$	+	$2$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$
					-	$7 t^{2}$	+	$7 t$
							+	$2 t$	-	$2$

Step 5.13

Multiply the new quotient term by the divisor.

				$4 t^{2}$	+	$7 t$	+	$2$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$
					-	$7 t^{2}$	+	$7 t$
							+	$2 t$	-	$2$
							+	$2 t$	-	$2$

Step 5.14

The expression needs to be subtracted from the dividend, so change all the signs in $2 t - 2$

				$4 t^{2}$	+	$7 t$	+	$2$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$
					-	$7 t^{2}$	+	$7 t$
							+	$2 t$	-	$2$
							-	$2 t$	+	$2$

Step 5.15

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

				$4 t^{2}$	+	$7 t$	+	$2$
$t$	-	$1$		$4 t^{3}$	+	$3 t^{2}$	-	$5 t$	-	$2$
			-	$4 t^{3}$	+	$4 t^{2}$
					+	$7 t^{2}$	-	$5 t$
					-	$7 t^{2}$	+	$7 t$
							+	$2 t$	-	$2$
							-	$2 t$	+	$2$
										$0$

Step 5.16

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

$4 t^{2} + 7 t + 2$

Step 6

Write $4 t^{3} + 3 t^{2} - 5 t - 2$ as a set of factors.

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