Examples

$10 x^{2} + k x + 8$

Step 1

Find the values of $a$ and $c$ in the trinomial $10 x^{2} + k x + 8$ with the format $a x^{2} + k x + c$ .

$a = 10$

$c = 8$

Step 2

For the trinomial $10 x^{2} + k x + 8$ , find the value of $a \cdot c$ .

$a \cdot c = 80$

Step 3

To find all possible values of $k$ , first find the factors of $a \cdot c$ $80$ . Once a factor is found, add it to its corresponding factor to get a possible value for $k$ . The factors for $80$ are all numbers between $- 80$ and $80$ , which divide $80$ evenly.

Check numbers between $- 80$ and $80$

Step 4

Calculate the factors of $80$ . Add corresponding factors to get all possible $k$ values.

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Step 4.1

Since $80$ divided by $- 80$ is the whole number $- 1$ , $- 80$ and $- 1$ are factors of $80$ .

$- 80$ and $- 1$ are factors

Step 4.2

Add the factors $- 80$ and $- 1$ together. Add $- 81$ to the list of possible $k$ values.

$k = - 81$

Step 4.3

Since $80$ divided by $- 40$ is the whole number $- 2$ , $- 40$ and $- 2$ are factors of $80$ .

$- 40$ and $- 2$ are factors

Step 4.4

Add the factors $- 40$ and $- 2$ together. Add $- 42$ to the list of possible $k$ values.

$k = - 81, - 42$

Step 4.5

Since $80$ divided by $- 20$ is the whole number $- 4$ , $- 20$ and $- 4$ are factors of $80$ .

$- 20$ and $- 4$ are factors

Step 4.6

Add the factors $- 20$ and $- 4$ together. Add $- 24$ to the list of possible $k$ values.

$k = - 81, - 42, - 24$

Step 4.7

Since $80$ divided by $- 16$ is the whole number $- 5$ , $- 16$ and $- 5$ are factors of $80$ .

$- 16$ and $- 5$ are factors

Step 4.8

Add the factors $- 16$ and $- 5$ together. Add $- 21$ to the list of possible $k$ values.

$k = - 81, - 42, - 24, - 21$

Step 4.9

Since $80$ divided by $- 10$ is the whole number $- 8$ , $- 10$ and $- 8$ are factors of $80$ .

$- 10$ and $- 8$ are factors

Step 4.10

Add the factors $- 10$ and $- 8$ together. Add $- 18$ to the list of possible $k$ values.

$k = - 81, - 42, - 24, - 21, - 18$

Step 4.11

Since $80$ divided by $1$ is the whole number $80$ , $1$ and $80$ are factors of $80$ .

$1$ and $80$ are factors

Step 4.12

Add the factors $1$ and $80$ together. Add $81$ to the list of possible $k$ values.

$k = - 81, - 42, - 24, - 21, - 18, 81$

Step 4.13

Since $80$ divided by $2$ is the whole number $40$ , $2$ and $40$ are factors of $80$ .

$2$ and $40$ are factors

Step 4.14

Add the factors $2$ and $40$ together. Add $42$ to the list of possible $k$ values.

$k = - 81, - 42, - 24, - 21, - 18, 81, 42$

Step 4.15

Since $80$ divided by $4$ is the whole number $20$ , $4$ and $20$ are factors of $80$ .

$4$ and $20$ are factors

Step 4.16

Add the factors $4$ and $20$ together. Add $24$ to the list of possible $k$ values.

$k = - 81, - 42, - 24, - 21, - 18, 81, 42, 24$

Step 4.17

Since $80$ divided by $5$ is the whole number $16$ , $5$ and $16$ are factors of $80$ .

$5$ and $16$ are factors

Step 4.18

Add the factors $5$ and $16$ together. Add $21$ to the list of possible $k$ values.

$k = - 81, - 42, - 24, - 21, - 18, 81, 42, 24, 21$

Step 4.19

Since $80$ divided by $8$ is the whole number $10$ , $8$ and $10$ are factors of $80$ .

$8$ and $10$ are factors

Step 4.20

Add the factors $8$ and $10$ together. Add $18$ to the list of possible $k$ values.

$k = - 81, - 42, - 24, - 21, - 18, 81, 42, 24, 21, 18$

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