Question

solve $2x^{3}+16x^{2}+30x = 0$. the roots are $x=square$, $x=square$, and $x=square$.

Explanation:

Step1: Factor out the GCF

First, we factor out the greatest common factor (GCF) from the polynomial \(2x^{3}+16x^{2}+30x\). The GCF of \(2x^{3}\), \(16x^{2}\), and \(30x\) is \(2x\). So we have:
\(2x(x^{2} + 8x + 15)=0\)

Step2: Factor the quadratic

Next, we factor the quadratic expression \(x^{2}+8x + 15\). We need two numbers that multiply to \(15\) and add up to \(8\). The numbers \(3\) and \(5\) satisfy this, so we can factor the quadratic as \((x + 3)(x + 5)\). Now our equation becomes:
\(2x(x + 3)(x + 5)=0\)

Step3: Apply the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\) (or both). So we set each factor equal to zero:

• For \(2x=0\), dividing both sides by \(2\) gives \(x = 0\).
• For \(x + 3=0\), subtracting \(3\) from both sides gives \(x=-3\).
• For \(x + 5=0\), subtracting \(5\) from both sides gives \(x=-5\).

Answer:

The roots are \(x = 0\), \(x=-3\), and \(x=-5\).