Precalculus Examples

$f (x) = {(x - 6)}^{2} {(x + 2)}^{2}$

Step 1

Simplify and reorder the polynomial in descending order to use Descartes rule.

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Step 1.1

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$(x - 6) (x - 6) {(x + 2)}^{2}$

Step 1.2

Expand $(x - 6) (x - 6)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$(x (x - 6) - 6 (x - 6)) {(x + 2)}^{2}$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $- 6$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $- 6$ by $- 6$ .

$(x^{2} - 6 x - 6 x + 36) {(x + 2)}^{2}$

Step 1.3.2

Subtract $6 x$ from $- 6 x$ .

$(x^{2} - 12 x + 36) {(x + 2)}^{2}$

Step 1.4

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$(x^{2} - 12 x + 36) ((x + 2) (x + 2))$

Step 1.5

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$(x^{2} - 12 x + 36) (x (x + 2) + 2 (x + 2))$

Step 1.5.2

Apply the distributive property.

$(x^{2} - 12 x + 36) (x \cdot x + x \cdot 2 + 2 (x + 2))$

Step 1.5.3

Apply the distributive property.

$(x^{2} - 12 x + 36) (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2)$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $x$ by $x$ .

$(x^{2} - 12 x + 36) (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2)$

Step 1.6.1.2

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

$(x^{2} - 12 x + 36) (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2)$

Step 1.6.1.3

Multiply $2$ by $2$ .

$(x^{2} - 12 x + 36) (x^{2} + 2 x + 2 x + 4)$

Step 1.6.2

Add $2 x$ and $2 x$ .

$(x^{2} - 12 x + 36) (x^{2} + 4 x + 4)$

Step 1.7

Expand $(x^{2} - 12 x + 36) (x^{2} + 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x^{2} + x^{2} (4 x) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8

Simplify terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.8.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 2} + x^{2} (4 x) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} (4 x) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.2

Rewrite using the commutative property of multiplication.

$x^{4} + 4 x^{2} x + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.8.1.3.1

Move $x$ .

$x^{4} + 4 (x \cdot x^{2}) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.8.1.3.2.1

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

$x^{4} + 4 (x^{1} x^{2}) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + 4 x^{1 + 2} + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.3.3

Add $1$ and $2$ .

$x^{4} + 4 x^{3} + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.4

Move $4$ to the left of $x^{2}$ .

$x^{4} + 4 x^{3} + 4 \cdot x^{2} - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.8.1.5.1

Move $x^{2}$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 (x^{2} x) - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.5.2

Multiply $x^{2}$ by $x$ .

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Step 1.8.1.5.2.1

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

$x^{4} + 4 x^{3} + 4 x^{2} - 12 (x^{2} x^{1}) - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{2 + 1} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.5.3

Add $2$ and $1$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.6

Rewrite using the commutative property of multiplication.

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 12 \cdot 4 x \cdot x - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.7

Multiply $x$ by $x$ by adding the exponents.

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Step 1.8.1.7.1

Move $x$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 12 \cdot 4 (x \cdot x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.7.2

Multiply $x$ by $x$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 12 \cdot 4 x^{2} - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.8

Multiply $- 12$ by $4$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 48 x^{2} - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.9

Multiply $4$ by $- 12$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 48 x^{2} - 48 x + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Step 1.8.1.10

Multiply $4$ by $36$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 48 x^{2} - 48 x + 36 x^{2} + 144 x + 36 \cdot 4$

Step 1.8.1.11

Multiply $36$ by $4$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 48 x^{2} - 48 x + 36 x^{2} + 144 x + 144$

Step 1.8.2

Simplify by adding terms.

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Step 1.8.2.1

Subtract $12 x^{3}$ from $4 x^{3}$ .

$x^{4} - 8 x^{3} + 4 x^{2} - 48 x^{2} - 48 x + 36 x^{2} + 144 x + 144$

Step 1.8.2.2

Subtract $48 x^{2}$ from $4 x^{2}$ .

$x^{4} - 8 x^{3} - 44 x^{2} - 48 x + 36 x^{2} + 144 x + 144$

Step 1.8.2.3

Add $- 44 x^{2}$ and $36 x^{2}$ .

$x^{4} - 8 x^{3} - 8 x^{2} - 48 x + 144 x + 144$

Step 1.8.2.4

Add $- 48 x$ and $144 x$ .

$x^{4} - 8 x^{3} - 8 x^{2} + 96 x + 144$

Step 2

To find the possible number of positive roots, look at the signs on the coefficients and count the number of times the signs on the coefficients change from positive to negative or negative to positive.

$f (x) = x^{4} - 8 x^{3} - 8 x^{2} + 96 x + 144$

Step 3

Since there are $2$ sign changes from the highest order term to the lowest, there are at most $2$ positive roots (Descartes' Rule of Signs). The other possible numbers of positive roots are found by subtracting off pairs of roots $(2 - 2)$ .

Positive Roots: $2$ or $0$

Step 4

To find the possible number of negative roots, replace $x$ with $- x$ and repeat the sign comparison.

$f (- x) = {(- x)}^{4} - 8 {(- x)}^{3} - 8 {(- x)}^{2} + 96 (- x) + 144$

Step 5

Simplify each term.

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Step 5.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{4} x^{4} - 8 {(- x)}^{3} - 8 {(- x)}^{2} + 96 (- x) + 144$

Step 5.2

Raise $- 1$ to the power of $4$ .

$f (- x) = 1 x^{4} - 8 {(- x)}^{3} - 8 {(- x)}^{2} + 96 (- x) + 144$

Step 5.3

Multiply $x^{4}$ by $1$ .

$f (- x) = x^{4} - 8 {(- x)}^{3} - 8 {(- x)}^{2} + 96 (- x) + 144$

Step 5.4

Apply the product rule to $- x$ .

$f (- x) = x^{4} - 8 ({(- 1)}^{3} x^{3}) - 8 {(- x)}^{2} + 96 (- x) + 144$

Step 5.5

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

$f (- x) = x^{4} - 8 (- x^{3}) - 8 {(- x)}^{2} + 96 (- x) + 144$

Step 5.6

Multiply $- 1$ by $- 8$ .

$f (- x) = x^{4} + 8 x^{3} - 8 {(- x)}^{2} + 96 (- x) + 144$

Step 5.7

Apply the product rule to $- x$ .

$f (- x) = x^{4} + 8 x^{3} - 8 ({(- 1)}^{2} x^{2}) + 96 (- x) + 144$

Step 5.8

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

$f (- x) = x^{4} + 8 x^{3} - 8 (1 x^{2}) + 96 (- x) + 144$

Step 5.9

Multiply $x^{2}$ by $1$ .

$f (- x) = x^{4} + 8 x^{3} - 8 x^{2} + 96 (- x) + 144$

Step 5.10

Multiply $- 1$ by $96$ .

$f (- x) = x^{4} + 8 x^{3} - 8 x^{2} - 96 x + 144$

Step 6

Since there are $2$ sign changes from the highest order term to the lowest, there are at most $2$ negative roots (Descartes' Rule of Signs). The other possible numbers of negative roots are found by subtracting off pairs of roots (e.g. $2 - 2$ ).

Negative Roots: $2$ or $0$

Step 7

The possible number of positive roots is $2$ or $0$ , and the possible number of negative roots is $2$ or $0$ .

Positive Roots: $2$ or $0$

Negative Roots: $2$ or $0$