Finite Math Examples

$f (- 3) = - 106$ , $f (- 1) = - 4$ , $f (1) = 2$ , $f (2) = 14$ , $f (x) = a x^{3} + b x^{2} + c x + d$

Step 1

Evaluate $f (- 3) = - 106$ .

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Step 1.1

Evaluate $f (- 3)$ .

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Step 1.1.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = a {(- 3)}^{3} + b {(- 3)}^{2} + c (- 3) + d$

Step 1.1.2

Simplify $a {(- 3)}^{3} + b {(- 3)}^{2} + c (- 3) + d$ .

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Step 1.1.2.1

Remove parentheses.

$a {(- 3)}^{3} + b {(- 3)}^{2} + c (- 3) + d$

Step 1.1.2.2

Simplify each term.

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Step 1.1.2.2.1

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

$a \cdot - 27 + b {(- 3)}^{2} + c (- 3) + d$

Step 1.1.2.2.2

Move $- 27$ to the left of $a$ .

$- 27 \cdot a + b {(- 3)}^{2} + c (- 3) + d$

Step 1.1.2.2.3

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

$- 27 a + b \cdot 9 + c (- 3) + d$

Step 1.1.2.2.4

Move $9$ to the left of $b$ .

$- 27 a + 9 \cdot b + c (- 3) + d$

Step 1.1.2.2.5

Move $- 3$ to the left of $c$ .

$- 27 a + 9 b - 3 c + d$

Step 1.2

Solve $- 27 a + 9 b - 3 c + d = - 106$ .

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Step 1.2.1

Move all terms not containing $a$ to the right side of the equation.

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Step 1.2.1.1

Subtract $9 b$ from both sides of the equation.

$- 27 a - 3 c + d = - 106 - 9 b$

Step 1.2.1.2

Add $3 c$ to both sides of the equation.

$- 27 a + d = - 106 - 9 b + 3 c$

Step 1.2.1.3

Subtract $d$ from both sides of the equation.

$- 27 a = - 106 - 9 b + 3 c - d$

Step 1.2.2

Divide each term in $- 27 a = - 106 - 9 b + 3 c - d$ by $- 27$ and simplify.

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Step 1.2.2.1

Divide each term in $- 27 a = - 106 - 9 b + 3 c - d$ by $- 27$ .

$\frac{- 27 a}{- 27} = \frac{- 106}{- 27} + \frac{- 9 b}{- 27} + \frac{3 c}{- 27} + \frac{- d}{- 27}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 27$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 27 a}{- 27} = \frac{- 106}{- 27} + \frac{- 9 b}{- 27} + \frac{3 c}{- 27} + \frac{- d}{- 27}$

Step 1.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 106}{- 27} + \frac{- 9 b}{- 27} + \frac{3 c}{- 27} + \frac{- d}{- 27}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Simplify each term.

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Step 1.2.2.3.1.1

Dividing two negative values results in a positive value.

$a = \frac{106}{27} + \frac{- 9 b}{- 27} + \frac{3 c}{- 27} + \frac{- d}{- 27}$

Step 1.2.2.3.1.2

Cancel the common factor of $- 9$ and $- 27$ .

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Step 1.2.2.3.1.2.1

Factor $- 9$ out of $- 9 b$ .

$a = \frac{106}{27} + \frac{- 9 (b)}{- 27} + \frac{3 c}{- 27} + \frac{- d}{- 27}$

Step 1.2.2.3.1.2.2

Cancel the common factors.

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Step 1.2.2.3.1.2.2.1

Factor $- 9$ out of $- 27$ .

$a = \frac{106}{27} + \frac{- 9 (b)}{- 9 (3)} + \frac{3 c}{- 27} + \frac{- d}{- 27}$

Step 1.2.2.3.1.2.2.2

Cancel the common factor.

$a = \frac{106}{27} + \frac{- 9 b}{- 9 \cdot 3} + \frac{3 c}{- 27} + \frac{- d}{- 27}$

Step 1.2.2.3.1.2.2.3

Rewrite the expression.

$a = \frac{106}{27} + \frac{b}{3} + \frac{3 c}{- 27} + \frac{- d}{- 27}$

Step 1.2.2.3.1.3

Cancel the common factor of $3$ and $- 27$ .

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Step 1.2.2.3.1.3.1

Factor $3$ out of $3 c$ .

$a = \frac{106}{27} + \frac{b}{3} + \frac{3 (c)}{- 27} + \frac{- d}{- 27}$

Step 1.2.2.3.1.3.2

Cancel the common factors.

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Step 1.2.2.3.1.3.2.1

Factor $3$ out of $- 27$ .

$a = \frac{106}{27} + \frac{b}{3} + \frac{3 c}{3 \cdot - 9} + \frac{- d}{- 27}$

Step 1.2.2.3.1.3.2.2

Cancel the common factor.

$a = \frac{106}{27} + \frac{b}{3} + \frac{3 c}{3 \cdot - 9} + \frac{- d}{- 27}$

Step 1.2.2.3.1.3.2.3

Rewrite the expression.

$a = \frac{106}{27} + \frac{b}{3} + \frac{c}{- 9} + \frac{- d}{- 27}$

Step 1.2.2.3.1.4

Move the negative in front of the fraction.

$a = \frac{106}{27} + \frac{b}{3} - \frac{c}{9} + \frac{- d}{- 27}$

Step 1.2.2.3.1.5

Dividing two negative values results in a positive value.

$a = \frac{106}{27} + \frac{b}{3} - \frac{c}{9} + \frac{d}{27}$

Step 2

Evaluate $f (- 1) = - 4$ .

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Step 2.1

Evaluate $f (- 1)$ .

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Step 2.1.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = a {(- 1)}^{3} + b {(- 1)}^{2} + c (- 1) + d$

Step 2.1.2

Simplify $a {(- 1)}^{3} + b {(- 1)}^{2} + c (- 1) + d$ .

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Step 2.1.2.1

Remove parentheses.

$a {(- 1)}^{3} + b {(- 1)}^{2} + c (- 1) + d$

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

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

$a \cdot - 1 + b {(- 1)}^{2} + c (- 1) + d$

Step 2.1.2.2.2

Move $- 1$ to the left of $a$ .

$- 1 \cdot a + b {(- 1)}^{2} + c (- 1) + d$

Step 2.1.2.2.3

Rewrite $- 1 a$ as $- a$ .

$- a + b {(- 1)}^{2} + c (- 1) + d$

Step 2.1.2.2.4

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

$- a + b \cdot 1 + c (- 1) + d$

Step 2.1.2.2.5

Multiply $b$ by $1$ .

$- a + b + c (- 1) + d$

Step 2.1.2.2.6

Move $- 1$ to the left of $c$ .

$- a + b - 1 c + d$

Step 2.1.2.2.7

Rewrite $- 1 c$ as $- c$ .

$- a + b - c + d$

Step 2.2

Solve $- a + b - c + d = - 4$ .

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Step 2.2.1

Move all terms not containing $a$ to the right side of the equation.

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Step 2.2.1.1

Subtract $b$ from both sides of the equation.

$- a - c + d = - 4 - b$

Step 2.2.1.2

Add $c$ to both sides of the equation.

$- a + d = - 4 - b + c$

Step 2.2.1.3

Subtract $d$ from both sides of the equation.

$- a = - 4 - b + c - d$

Step 2.2.2

Divide each term in $- a = - 4 - b + c - d$ by $- 1$ and simplify.

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Step 2.2.2.1

Divide each term in $- a = - 4 - b + c - d$ by $- 1$ .

$\frac{- a}{- 1} = \frac{- 4}{- 1} + \frac{- b}{- 1} + \frac{c}{- 1} + \frac{- d}{- 1}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{a}{1} = \frac{- 4}{- 1} + \frac{- b}{- 1} + \frac{c}{- 1} + \frac{- d}{- 1}$

Step 2.2.2.2.2

Divide $a$ by $1$ .

$a = \frac{- 4}{- 1} + \frac{- b}{- 1} + \frac{c}{- 1} + \frac{- d}{- 1}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Simplify each term.

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Step 2.2.2.3.1.1

Divide $- 4$ by $- 1$ .

$a = 4 + \frac{- b}{- 1} + \frac{c}{- 1} + \frac{- d}{- 1}$

Step 2.2.2.3.1.2

Dividing two negative values results in a positive value.

$a = 4 + \frac{b}{1} + \frac{c}{- 1} + \frac{- d}{- 1}$

Step 2.2.2.3.1.3

Divide $b$ by $1$ .

$a = 4 + b + \frac{c}{- 1} + \frac{- d}{- 1}$

Step 2.2.2.3.1.4

Move the negative one from the denominator of $\frac{c}{- 1}$ .

$a = 4 + b - 1 \cdot c + \frac{- d}{- 1}$

Step 2.2.2.3.1.5

Rewrite $- 1 \cdot c$ as $- c$ .

$a = 4 + b - c + \frac{- d}{- 1}$

Step 2.2.2.3.1.6

Dividing two negative values results in a positive value.

$a = 4 + b - c + \frac{d}{1}$

Step 2.2.2.3.1.7

Divide $d$ by $1$ .

$a = 4 + b - c + d$

Step 3

Evaluate $f (1) = 2$ .

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Step 3.1

Evaluate $f (1)$ .

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Step 3.1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = a {(1)}^{3} + b {(1)}^{2} + c (1) + d$

Step 3.1.2

Simplify $a {(1)}^{3} + b {(1)}^{2} + c (1) + d$ .

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Step 3.1.2.1

Remove parentheses.

$a {(1)}^{3} + b {(1)}^{2} + c (1) + d$

Step 3.1.2.2

Simplify each term.

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Step 3.1.2.2.1

One to any power is one.

$a \cdot 1 + b {(1)}^{2} + c (1) + d$

Step 3.1.2.2.2

Multiply $a$ by $1$ .

$a + b {(1)}^{2} + c (1) + d$

Step 3.1.2.2.3

One to any power is one.

$a + b \cdot 1 + c (1) + d$

Step 3.1.2.2.4

Multiply $b$ by $1$ .

$a + b + c (1) + d$

Step 3.1.2.2.5

Multiply $c$ by $1$ .

$a + b + c + d$

Step 3.2

Solve $a + b + c + d = 2$ .

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Step 3.2.1

Move all terms not containing $a$ to the right side of the equation.

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Step 3.2.1.1

Subtract $b$ from both sides of the equation.

$a + c + d = 2 - b$

Step 3.2.1.2

Subtract $c$ from both sides of the equation.

$a + d = 2 - b - c$

Step 3.2.1.3

Subtract $d$ from both sides of the equation.

$a = 2 - b - c - d$

Step 4

Evaluate $f (2) = 14$ .

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Step 4.1

Evaluate $f (2)$ .

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Step 4.1.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = a {(2)}^{3} + b {(2)}^{2} + c (2) + d$

Step 4.1.2

Simplify $a {(2)}^{3} + b {(2)}^{2} + c (2) + d$ .

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Step 4.1.2.1

Remove parentheses.

$a {(2)}^{3} + b {(2)}^{2} + c (2) + d$

Step 4.1.2.2

Simplify each term.

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Step 4.1.2.2.1

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

$a \cdot 8 + b {(2)}^{2} + c (2) + d$

Step 4.1.2.2.2

Move $8$ to the left of $a$ .

$8 \cdot a + b {(2)}^{2} + c (2) + d$

Step 4.1.2.2.3

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

$8 a + b \cdot 4 + c (2) + d$

Step 4.1.2.2.4

Move $4$ to the left of $b$ .

$8 a + 4 \cdot b + c (2) + d$

Step 4.1.2.2.5

Move $2$ to the left of $c$ .

$8 a + 4 b + 2 c + d$

Step 4.2

Solve $8 a + 4 b + 2 c + d = 14$ .

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Step 4.2.1

Move all terms not containing $a$ to the right side of the equation.

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Step 4.2.1.1

Subtract $4 b$ from both sides of the equation.

$8 a + 2 c + d = 14 - 4 b$

Step 4.2.1.2

Subtract $2 c$ from both sides of the equation.

$8 a + d = 14 - 4 b - 2 c$

Step 4.2.1.3

Subtract $d$ from both sides of the equation.

$8 a = 14 - 4 b - 2 c - d$

Step 4.2.2

Divide each term in $8 a = 14 - 4 b - 2 c - d$ by $8$ and simplify.

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Step 4.2.2.1

Divide each term in $8 a = 14 - 4 b - 2 c - d$ by $8$ .

$\frac{8 a}{8} = \frac{14}{8} + \frac{- 4 b}{8} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $8$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{8 a}{8} = \frac{14}{8} + \frac{- 4 b}{8} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{14}{8} + \frac{- 4 b}{8} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Simplify each term.

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Step 4.2.2.3.1.1

Cancel the common factor of $14$ and $8$ .

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Step 4.2.2.3.1.1.1

Factor $2$ out of $14$ .

$a = \frac{2 (7)}{8} + \frac{- 4 b}{8} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.1.2

Cancel the common factors.

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Step 4.2.2.3.1.1.2.1

Factor $2$ out of $8$ .

$a = \frac{2 \cdot 7}{2 \cdot 4} + \frac{- 4 b}{8} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.1.2.2

Cancel the common factor.

$a = \frac{2 \cdot 7}{2 \cdot 4} + \frac{- 4 b}{8} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.1.2.3

Rewrite the expression.

$a = \frac{7}{4} + \frac{- 4 b}{8} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.2

Cancel the common factor of $- 4$ and $8$ .

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Step 4.2.2.3.1.2.1

Factor $4$ out of $- 4 b$ .

$a = \frac{7}{4} + \frac{4 (- b)}{8} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.2.2

Cancel the common factors.

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Step 4.2.2.3.1.2.2.1

Factor $4$ out of $8$ .

$a = \frac{7}{4} + \frac{4 (- b)}{4 (2)} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.2.2.2

Cancel the common factor.

$a = \frac{7}{4} + \frac{4 (- b)}{4 \cdot 2} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.2.2.3

Rewrite the expression.

$a = \frac{7}{4} + \frac{- b}{2} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.3

Move the negative in front of the fraction.

$a = \frac{7}{4} - \frac{b}{2} + \frac{- 2 c}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.4

Cancel the common factor of $- 2$ and $8$ .

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Step 4.2.2.3.1.4.1

Factor $2$ out of $- 2 c$ .

$a = \frac{7}{4} - \frac{b}{2} + \frac{2 (- c)}{8} + \frac{- d}{8}$

Step 4.2.2.3.1.4.2

Cancel the common factors.

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Step 4.2.2.3.1.4.2.1

Factor $2$ out of $8$ .

$a = \frac{7}{4} - \frac{b}{2} + \frac{2 (- c)}{2 (4)} + \frac{- d}{8}$

Step 4.2.2.3.1.4.2.2

Cancel the common factor.

$a = \frac{7}{4} - \frac{b}{2} + \frac{2 (- c)}{2 \cdot 4} + \frac{- d}{8}$

Step 4.2.2.3.1.4.2.3

Rewrite the expression.

$a = \frac{7}{4} - \frac{b}{2} + \frac{- c}{4} + \frac{- d}{8}$

Step 4.2.2.3.1.5

Move the negative in front of the fraction.

$a = \frac{7}{4} - \frac{b}{2} - \frac{c}{4} + \frac{- d}{8}$

Step 4.2.2.3.1.6

Move the negative in front of the fraction.

$a = \frac{7}{4} - \frac{b}{2} - \frac{c}{4} - \frac{d}{8}$

Step 5

List all of the solutions.

$f (- 3) = - 106 = a = \frac{106}{27} + \frac{b}{3} - \frac{c}{9} + \frac{d}{27}$

$f (- 1) = - 4 = a = 4 + b - c + d$

$f (1) = 2 = a = 2 - b - c - d$

$f (2) = 14 = a = \frac{7}{4} - \frac{b}{2} - \frac{c}{4} - \frac{d}{8}$