Questions: Factor the following polynomial completely. 10 x^4 + 130 x^3 + 420 x^2 A. 10 x^2(x+6)(x+7) B. -10 x^2(x+6)(x+7) C. -10 x^2(x-6)(x+7) D. 10 x^2(x+6)(x-7)

Transcript text: Factor the following polynomial completely. \[ 10 x^{4}+130 x^{3}+420 x^{2} \] A. $10 x^{2}(x+6)(x+7)$ B. $-10 x^{2}(x+6)(x+7)$ C. $-10 x^{2}(x-6)(x+7)$ D. $10 x^{2}(x+6)(x-7)$

Solution

Solution Steps

To factor the given polynomial completely, we first look for the greatest common factor (GCF) of all the terms. Then, we factor the remaining polynomial by finding its roots or using other factoring techniques.

Step 1: Identify the Polynomial

We start with the polynomial given by \[ 10x^4 + 130x^3 + 420x^2. \]

Step 2: Factor Out the Greatest Common Factor

The greatest common factor (GCF) of the terms $10x^4$, $130x^3$, and $420x^2$ is $10x^2$. We factor this out: \[ 10x^2(x^2 + 13x + 42). \]

Step 3: Factor the Quadratic Expression

Next, we need to factor the quadratic expression $x^2 + 13x + 42$. We look for two numbers that multiply to $42$ and add to $13$. These numbers are $6$ and $7$. Thus, we can factor the quadratic as: \[ x^2 + 13x + 42 = (x + 6)(x + 7). \]

Step 4: Combine the Factors

Putting it all together, we have: \[ 10x^2(x + 6)(x + 7). \]