Find the Maximum Number of Real Roots f(x)=-2(x-4)(x+3)^2

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

Step 1

Simplify by multiplying through.

Apply the distributive property.

$f (x) = (- 2 x - 2 \cdot - 4) {(x + 3)}^{2}$

Simplify the expression.

Multiply $- 2$ by $- 4$ .

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

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$f (x) = (- 2 x + 8) ((x + 3) (x + 3))$

Step 2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$f (x) = (- 2 x + 8) (x (x + 3) + 3 (x + 3))$

Apply the distributive property.

$f (x) = (- 2 x + 8) (x \cdot x + x \cdot 3 + 3 (x + 3))$

Apply the distributive property.

$f (x) = (- 2 x + 8) (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = (- 2 x + 8) (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3)$

Move $3$ to the left of $x$ .

$f (x) = (- 2 x + 8) (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3)$

Multiply $3$ by $3$ .

$f (x) = (- 2 x + 8) (x^{2} + 3 x + 3 x + 9)$

Add $3 x$ and $3 x$ .

$f (x) = (- 2 x + 8) (x^{2} + 6 x + 9)$

Step 4

Expand $(- 2 x + 8) (x^{2} + 6 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = - 2 x \cdot x^{2} - 2 x (6 x) - 2 x \cdot 9 + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

Step 5

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$f (x) = - 2 (x^{2} x) - 2 x (6 x) - 2 x \cdot 9 + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

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

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

$f (x) = - 2 (x^{2} x) - 2 x (6 x) - 2 x \cdot 9 + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

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

$f (x) = - 2 x^{2 + 1} - 2 x (6 x) - 2 x \cdot 9 + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

Add $2$ and $1$ .

$f (x) = - 2 x^{3} - 2 x (6 x) - 2 x \cdot 9 + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

Rewrite using the commutative property of multiplication.

$f (x) = - 2 x^{3} - 2 \cdot (6 x \cdot x) - 2 x \cdot 9 + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

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

Move $x$ .

$f (x) = - 2 x^{3} - 2 \cdot (6 (x \cdot x)) - 2 x \cdot 9 + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

Multiply $x$ by $x$ .

$f (x) = - 2 x^{3} - 2 \cdot (6 x^{2}) - 2 x \cdot 9 + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

Multiply $- 2$ by $6$ .

$f (x) = - 2 x^{3} - 12 x^{2} - 2 x \cdot 9 + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

Multiply $9$ by $- 2$ .

$f (x) = - 2 x^{3} - 12 x^{2} - 18 x + 8 x^{2} + 8 (6 x) + 8 \cdot 9$

Multiply $6$ by $8$ .

$f (x) = - 2 x^{3} - 12 x^{2} - 18 x + 8 x^{2} + 48 x + 8 \cdot 9$

Multiply $8$ by $9$ .

$f (x) = - 2 x^{3} - 12 x^{2} - 18 x + 8 x^{2} + 48 x + 72$

Simplify by adding terms.

Add $- 12 x^{2}$ and $8 x^{2}$ .

$f (x) = - 2 x^{3} - 4 x^{2} - 18 x + 48 x + 72$

Add $- 18 x$ and $48 x$ .

$f (x) = - 2 x^{3} - 4 x^{2} + 30 x + 72$

Step 6

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$- 2 x^{3} \to 3$

$- 4 x^{2} \to 2$

$30 x \to 1$

$72 \to 0$

Step 7

The largest exponent is the degree of the polynomial.

$3$

Step 8

The maximum number of possible roots is the degree of $- 2 x^{3} - 4 x^{2} + 30 x + 72$ .

$3$

Find the Maximum Number of Real Roots f(x)=-2(x-4)(x+3)^2

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