One example of a polynomial that cannot be factored further is:
x^2 + 5
To understand why this polynomial cannot be factored further, we need to consider its terms and coefficients. In this case, we have two terms: x^2 and 5. Both of these terms are prime numbers, meaning they cannot be factored into smaller integer factors. Additionally, there is no common factor that can be factored out of both terms. Therefore, the polynomial x^2 + 5 is already in its simplest form and cannot be factored further using integer coefficients.
There are several factoring techniques that might be used with this polynomial, such as factoring by grouping, factoring by completing the square, or using the quadratic formula. However, none of these techniques will work in this case, because the polynomial x^2 + 5 is not factorable over the integers.
For example, if we try factoring by grouping, we might attempt to write:
x^2 + 5 = (x + a)(x + b)
where a and b are constants that we want to determine. However, if we expand the right-hand side of this equation, we get:
x^2 + (a + b)x + ab
This means that in order for this factoring to work, we need a and b to satisfy the equations:
a + b = 0
ab = 5
However, there are no integer values of a and b that satisfy these equations, because 5 is a prime number and cannot be factored into smaller integers. Therefore, factoring by grouping does not work for this polynomial.
Similarly, factoring by completing the square or using the quadratic formula also do not work for this polynomial, because the discriminant (b^2 - 4ac) is negative, which means that the roots of the polynomial are complex numbers, not real numbers. Therefore, x^2 + 5 cannot be factored further using any of these factoring techniques.