Questions: Factor the following by grouping. a^3 + 3a^2 + a + 3

Solution Steps

To factor the polynomial by grouping, we first split the polynomial into two groups. Then, we factor out the greatest common factor from each group. Finally, we look for a common binomial factor in the resulting expression and factor it out.

We start with the polynomial \( a^3 + 3a^2 + a + 3 \). We can group the terms as follows: \[ (a^3 + 3a^2) + (a + 3) \]

Next, we factor out the greatest common factor from each group: \[ a^2(a + 3) + 1(a + 3) \]

Now, we notice that \( (a + 3) \) is a common factor: \[ (a + 3)(a^2 + 1) \]

Final Answer