Question

1
select the correct answer.
one of the factors of the polynomial $x^3 + 5x^2 + 6x$ is $(x^2 + 3x)$. what is the other factor?
a. $x - 2$
b. $x + 2$
c. $x + 1$
d. $x - 1$

Explanation:

Step1: Factor out x from the polynomial

First, factor out the greatest common factor (GCF) from the polynomial \(x^3 + 5x^2 + 6x\). The GCF of the terms \(x^3\), \(5x^2\), and \(6x\) is \(x\). So, we can factor out \(x\) from each term:

$$ x^3 + 5x^2 + 6x = x(x^2 + 5x + 6) $$

Step2: Factor the quadratic expression

Now, we need to factor the quadratic expression \(x^2 + 5x + 6\). We look for two numbers that multiply to \(6\) (the constant term) and add up to \(5\) (the coefficient of the middle term). The numbers \(2\) and \(3\) satisfy this because \(2 \times 3 = 6\) and \(2 + 3 = 5\). So, we can factor \(x^2 + 5x + 6\) as \((x + 2)(x + 3)\). Therefore, the polynomial becomes:

$$ x(x^2 + 5x + 6) = x(x + 2)(x + 3) $$

Step3: Compare with the given factor

We are given that one of the factors is \((x^2 + 3x)\). Let's factor \((x^2 + 3x)\) to see its form:

$$ x^2 + 3x = x(x + 3) $$

Now, looking at the factored form of the polynomial \(x(x + 2)(x + 3)\), if we group \(x(x + 3)\) (which is \((x^2 + 3x)\)) as one factor, the remaining factor is \((x + 2)\).

Answer:

B. \(x + 2\)