Algebra Examples

$g (x) = y - 7$ , $f (x) = x^{3} - 9 x^{2} - 34 x + 336$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (y - 7)$ by substituting in the value of $g$ into $f$ .

$f (y - 7) = {(y - 7)}^{3} - 9 {(y - 7)}^{2} - 34 (y - 7) + 336$

Step 3

Simplify each term.

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Step 3.1

Use the Binomial Theorem.

$f (y - 7) = y^{3} + 3 y^{2} \cdot - 7 + 3 y {(- 7)}^{2} + {(- 7)}^{3} - 9 {(y - 7)}^{2} - 34 (y - 7) + 336$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $- 7$ by $3$ .

$f (y - 7) = y^{3} - 21 y^{2} + 3 y {(- 7)}^{2} + {(- 7)}^{3} - 9 {(y - 7)}^{2} - 34 (y - 7) + 336$

Step 3.2.2

Raise $- 7$ to the power of $2$ .

$f (y - 7) = y^{3} - 21 y^{2} + 3 y \cdot 49 + {(- 7)}^{3} - 9 {(y - 7)}^{2} - 34 (y - 7) + 336$

Step 3.2.3

Multiply $49$ by $3$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y + {(- 7)}^{3} - 9 {(y - 7)}^{2} - 34 (y - 7) + 336$

Step 3.2.4

Raise $- 7$ to the power of $3$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 {(y - 7)}^{2} - 34 (y - 7) + 336$

Step 3.3

Rewrite ${(y - 7)}^{2}$ as $(y - 7) (y - 7)$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 ((y - 7) (y - 7)) - 34 (y - 7) + 336$

Step 3.4

Expand $(y - 7) (y - 7)$ using the FOIL Method.

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Step 3.4.1

Apply the distributive property.

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 (y (y - 7) - 7 (y - 7)) - 34 (y - 7) + 336$

Step 3.4.2

Apply the distributive property.

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 (y \cdot y + y \cdot - 7 - 7 (y - 7)) - 34 (y - 7) + 336$

Step 3.4.3

Apply the distributive property.

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 (y \cdot y + y \cdot - 7 - 7 y - 7 \cdot - 7) - 34 (y - 7) + 336$

Step 3.5

Simplify and combine like terms.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply $y$ by $y$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 (y^{2} + y \cdot - 7 - 7 y - 7 \cdot - 7) - 34 (y - 7) + 336$

Step 3.5.1.2

Move $- 7$ to the left of $y$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 (y^{2} - 7 \cdot y - 7 y - 7 \cdot - 7) - 34 (y - 7) + 336$

Step 3.5.1.3

Multiply $- 7$ by $- 7$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 (y^{2} - 7 y - 7 y + 49) - 34 (y - 7) + 336$

Step 3.5.2

Subtract $7 y$ from $- 7 y$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 (y^{2} - 14 y + 49) - 34 (y - 7) + 336$

Step 3.6

Apply the distributive property.

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 y^{2} - 9 (- 14 y) - 9 \cdot 49 - 34 (y - 7) + 336$

Step 3.7

Simplify.

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Step 3.7.1

Multiply $- 14$ by $- 9$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 y^{2} + 126 y - 9 \cdot 49 - 34 (y - 7) + 336$

Step 3.7.2

Multiply $- 9$ by $49$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 y^{2} + 126 y - 441 - 34 (y - 7) + 336$

Step 3.8

Apply the distributive property.

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 y^{2} + 126 y - 441 - 34 y - 34 \cdot - 7 + 336$

Step 3.9

Multiply $- 34$ by $- 7$ .

$f (y - 7) = y^{3} - 21 y^{2} + 147 y - 343 - 9 y^{2} + 126 y - 441 - 34 y + 238 + 336$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $9 y^{2}$ from $- 21 y^{2}$ .

$f (y - 7) = y^{3} - 30 y^{2} + 147 y - 343 + 126 y - 441 - 34 y + 238 + 336$

Step 4.2

Add $147 y$ and $126 y$ .

$f (y - 7) = y^{3} - 30 y^{2} + 273 y - 343 - 441 - 34 y + 238 + 336$

Step 4.3

Subtract $34 y$ from $273 y$ .

$f (y - 7) = y^{3} - 30 y^{2} + 239 y - 343 - 441 + 238 + 336$

Step 4.4

Simplify by adding and subtracting.

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Step 4.4.1

Subtract $441$ from $- 343$ .

$f (y - 7) = y^{3} - 30 y^{2} + 239 y - 784 + 238 + 336$

Step 4.4.2

Add $- 784$ and $238$ .

$f (y - 7) = y^{3} - 30 y^{2} + 239 y - 546 + 336$

Step 4.4.3

Add $- 546$ and $336$ .

$f (y - 7) = y^{3} - 30 y^{2} + 239 y - 210$