Factor the polynomial completely. Must show work/steps on your work page3⁢𝑥3+19⁢𝑥2−40⁢𝑥 =

Sure, let's factor the polynomial $3x^3 + 19x^2 - 40x$ step by step.

Step 1: Factor out the Greatest Common Factor (GCF)

First, we look for the greatest common factor (GCF) of all the terms in the polynomial. In this case, each term has an $x$ in common.

$3x^3 + 19x^2 - 40x = x(3x^2 + 19x - 40)$

Step 2: Factor the Quadratic Expression

Next, we need to factor the quadratic expression $3x^2 + 19x - 40$. We will use the method of factoring by grouping.

Step 2.1: Find two numbers that multiply to $a \cdot c$ and add to $b$

For the quadratic $ax^2 + bx + c$, we need to find two numbers that multiply to $a \cdot c$ and add to $b$.

Here, $a = 3$, $b = 19$, and $c = -40$.

$a \cdot c = 3 \cdot (-40) = -120$ We need two numbers that multiply to $-120$ and add to $19$. These numbers are $24$ and $-5$.

Step 2.2: Rewrite the middle term using these two numbers

Rewrite $19x$ as $24x - 5x$:

$3x^2 + 19x - 40 = 3x^2 + 24x - 5x - 40$

Step 2.3: Factor by grouping

Group the terms in pairs and factor out the common factors from each pair:

$3x^2 + 24x - 5x - 40 = 3x(x + 8) - 5(x + 8)$

Step 2.4: Factor out the common binomial factor

Now, factor out the common binomial factor $(x + 8)$:

$3x(x + 8) - 5(x + 8) = (3x - 5)(x + 8)$

Step 3: Combine the factors

Now, we combine the factors we have found:

$x(3x^2 + 19x - 40) = x(3x - 5)(x + 8)$

Final Answer

The completely factored form of the polynomial $3x^3 + 19x^2 - 40x$ is:

$3x^3 + 19x^2 - 40x = x(3x - 5)(x + 8)$