Identify the Zeros and Their Multiplicities f(x)=2x^2-7x-15

$f (x) = 2 x^{2} - 7 x - 15$

Step 1

Set $2 x^{2} - 7 x - 15$ equal to $0$ .

$2 x^{2} - 7 x - 15 = 0$

Step 2

Solve for $x$ .

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 x$ .

$2 x^{2} - 7 x - 15 = 0$

Rewrite $- 7$ as $3$ plus $- 10$

$2 x^{2} + (3 - 10) x - 15 = 0$

Apply the distributive property.

$2 x^{2} + 3 x - 10 x - 15 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} + 3 x) - 10 x - 15 = 0$

Factor out the greatest common factor (GCF) from each group.

$x (2 x + 3) - 5 (2 x + 3) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$(2 x + 3) (x - 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x + 3 = 0$

$x - 5 = 0$

Set $2 x + 3$ equal to $0$ and solve for $x$ .

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Solve $2 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Divide each term in $2 x = - 3$ by $2$ and simplify.

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(2 x + 3) (x - 5) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - \frac{3}{2}$ (Multiplicity of $1$ )

$x = 5$ (Multiplicity of $1$ )

$x = - \frac{3}{2}$ (Multiplicity of $1$ )

$x = 5$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=2x^2-7x-15

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