Video Transcript
Factorize fully six 𝑥 cubed minus 17𝑥 squared plus 12𝑥.
In this question, we are asked to fully factor an algebraic expression. And we can start by noting that there is only a single variable of 𝑥 and the powers of 𝑥 that appear in the expression are three, two, and one. So this is a single-variable cubic polynomial.
The first thing that we should always check is if the terms of our expression share a common factor that we can start by taking out. We should check the coefficients and the variables. Let’s start with the coefficients. We note that 17 is prime. And so we can find that the highest common factor of the coefficients is one. If we check the terms for shared factors of 𝑥, we can see that all three terms do have a factor of 𝑥. We can take this shared factor of 𝑥 out of the terms by reducing the powers of 𝑥 in each term by one. We obtain 𝑥 multiplied by six 𝑥 squared minus 17𝑥 plus 12.
Since we need to fully factor the expression, we now need to attempt to factor the quadratic factor. To do this, we first note that the leading coefficient is not one. In order to factor a nonmonic quadratic, say 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, we need to follow the following steps. The first thing that we need to do is list all of the factor pairs of 𝑎 times 𝑐. If we do this, then we obtain the following factor pairs of 72, where we note that we can also take the negative of each pair as well.
Now since 𝑎 and 𝑐 have the same sign, we need to find a factor pair with the same sign that add to give the value of 𝑏, which is negative 17. By checking the factor pairs, we can see that negative eight plus negative nine is equal to negative 17.
We now need to rewrite the 𝑥-term using this factor pair and then apply factoring by grouping. We can rewrite our expression as 𝑥 multiplied by six 𝑥 squared minus nine 𝑥 minus eight 𝑥 plus 12.
We now need to factor each pair of terms in the quadratic. Let’s start with the first pair, we see that they share a factor of three 𝑥. So we take out the factor of three 𝑥 from these two terms to get three 𝑥 times two 𝑥 minus three. We then want to take out a shared factor from the second two terms to find a shared factor with our other terms. In this case, we take out a factor of negative four. This gives us negative four times two 𝑥 minus three. And we can now see that there is a shared factor of two 𝑥 minus three that we can take out.
Taking out this linear factor then gives us 𝑥 times three 𝑥 minus four times two 𝑥 minus three. Finally, we can note that our factors are linear and there are no common factors to take out, so we cannot factor any further. Hence, we have fully factored the expression to obtain 𝑥 times three 𝑥 minus four times two 𝑥 minus three.
