Question
Use factoring to solve the polynomial equation. \[ 2 x^{3}+10 x^{2}+12 x=0 \]
Ask by Cross Luna. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are \(x=-3\), \(x=-2\), and \(x=0\).
Solution
Solve the equation \( 2x^3+10x^2+12x=0 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(2x^{3}+10x^{2}+12x=0\)
- step1: Factor the expression:
\(2x\left(x+2\right)\left(x+3\right)=0\)
- step2: Divide both sides:
\(x\left(x+2\right)\left(x+3\right)=0\)
- step3: Separate into possible cases:
\(\begin{align}&x=0\\&x+2=0\\&x+3=0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=0\\&x=-2\\&x=-3\end{align}\)
- step5: Rewrite:
\(x_{1}=-3,x_{2}=-2,x_{3}=0\)
The solutions to the polynomial equation \(2x^{3}+10x^{2}+12x=0\) are \(x=-3\), \(x=-2\), and \(x=0\).
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Simplify this solution The Deep Dive
To solve the polynomial equation \( 2x^{3} + 10x^{2} + 12x = 0 \) using factoring, we start by factoring out the greatest common factor, which in this case is \( 2x \): \[ 2x(x^{2} + 5x + 6) = 0. \] Next, we need to factor the quadratic \( x^{2} + 5x + 6 \). This can be factored as: \[ (x + 2)(x + 3). \] Putting it all together, we have: \[ 2x(x + 2)(x + 3) = 0. \] Now, we can set each factor to zero: 1. \( 2x = 0 \) gives \( x = 0 \). 2. \( x + 2 = 0 \) gives \( x = -2 \). 3. \( x + 3 = 0 \) gives \( x = -3 \). Thus, the solutions to the equation are: \[ x = 0, \, x = -2, \, x = -3. \]
