Determine whether the following expression is prime or not: 𝑥 squared plus five 𝑥 plus five. If it is not prime, give the expression in its factorized form.
In order to determine whether the quadratic is prime or not, we need to decide if it can be factorized. Any quadratic is prime if it cannot be factorized. The quadratic in this example, 𝑥 squared plus five 𝑥 plus five, has a leading coefficient or coefficient of 𝑥 squared equal to one.
This means that it is a case of simple factorization, where we need to find two numbers with a product of plus five and a sum of plus five. The sum of the two numbers needs to be the coefficient of 𝑥. And the product of the two numbers needs to be equal to the free term. The only product of two integers that equals five is one and five, as one multiplied by five is equal to five.
However, these numbers do not have a sum of five, as one plus five is equal to six, not five. As there is no pair of numbers with a product of plus five and a sum of plus five, we can say that the quadratic 𝑥 squared plus five 𝑥 plus five is prime and therefore cannot be factorized.