Finite Math Examples

x^{2} - 6 x + 9

$x^{2} - 6 x + 9$

Step 1

If a polynomial function has integer coefficients, then every rational zero will have the form

\frac{p}{q}

$\frac{p}{q}$ where

p

$p$ is a factor of the constant and

q

$q$ is a factor of the leading coefficient.

p = \pm 1, \pm 3, \pm 9

$p = \pm 1, \pm 3, \pm 9$

q = \pm 1

$q = \pm 1$

Step 2

Find every combination of

\pm \frac{p}{q}

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

\pm 1, \pm 3, \pm 9

$\pm 1, \pm 3, \pm 9$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is

0

$0$ , which means it is a root.

{(3)}^{2} - 6 \cdot 3 + 9

${(3)}^{2} - 6 \cdot 3 + 9$

Step 4

Simplify the expression. In this case, the expression is equal to

0

$0$ so

x = 3

$x = 3$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise

3

$3$ to the power of

2

$2$ .

9 - 6 \cdot 3 + 9

$9 - 6 \cdot 3 + 9$

Step 4.1.2

Multiply

- 6

$- 6$ by

3

$3$ .

9 - 18 + 9

$9 - 18 + 9$

9 - 18 + 9

$9 - 18 + 9$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract

18

$18$ from

9

$9$ .

- 9 + 9

$- 9 + 9$

Step 4.2.2

Add

- 9

$- 9$ and

9

$9$ .

0

$0$

0

$0$

0

$0$

Step 5

Since

3

$3$ is a known root, divide the polynomial by

x - 3

$x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

\frac{x^{2} - 6 x + 9}{x - 3}

$\frac{x^{2} - 6 x + 9}{x - 3}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by

1

$1$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$3$ $3$	$1$ $1$	$- 6$ $- 6$	$9$ $9$

Step 6.2

The first number in the dividend

(1)

$(1)$ is put into the first position of the result area (below the horizontal line).

$3$ $3$	$1$ $1$	$- 6$ $- 6$	$9$ $9$

	$1$ $1$

Step 6.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(3)

$(3)$ and place the result of

(3)

$(3)$ under the next term in the dividend

(- 6)

$(- 6)$ .

$3$ $3$	$1$ $1$	$- 6$ $- 6$	$9$ $9$
		$3$ $3$
	$1$ $1$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$ $3$	$1$ $1$	$- 6$ $- 6$	$9$ $9$
		$3$ $3$
	$1$ $1$	$- 3$ $- 3$

Step 6.5

Multiply the newest entry in the result

(- 3)

$(- 3)$ by the divisor

(3)

$(3)$ and place the result of

(- 9)

$(- 9)$ under the next term in the dividend

(9)

$(9)$ .

$3$ $3$	$1$ $1$	$- 6$ $- 6$	$9$ $9$
		$3$ $3$	$- 9$ $- 9$
	$1$ $1$	$- 3$ $- 3$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$ $3$	$1$ $1$	$- 6$ $- 6$	$9$ $9$
		$3$ $3$	$- 9$ $- 9$
	$1$ $1$	$- 3$ $- 3$	$0$ $0$

Step 6.7

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

(1) x - 3

$(1) x - 3$

Step 6.8

Simplify the quotient polynomial.

x - 3

$x - 3$

x - 3

$x - 3$

Step 7

Add

3

$3$ to both sides of the equation.

x = 3

$x = 3$

Step 8

The polynomial can be written as a set of linear factors.

x - 3

$x - 3$

Step 9

These are the roots (zeros) of the polynomial

x^{2} - 6 x + 9

$x^{2} - 6 x + 9$ .

x = 3

$x = 3$

Step 10

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