Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{2}+4 x+16 $$

Short answer

The polynomial is prime.

Step-by-step solution

Identify the Structure

The given polynomial is a quadratic of the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 4 \), and \( c = 16 \).

Calculate the Discriminant

Calculate the discriminant using the formula \( \text{Discriminant} = b^2 - 4ac \).Substitute the values: \( \text{Discriminant} = 4^2 - 4 \times 1 \times 16 = 16 - 64 = -48 \).

Analyze the Discriminant

Since the discriminant is negative (\(-48\)), the polynomial has no real roots and cannot be factored over the real numbers.

Conclusion

As the polynomial has no real roots, it cannot be factored using real numbers, and is therefore considered prime in the context of real numbers.

Key concepts

quadratic polynomial

A quadratic polynomial is a type of polynomial that has a degree of 2. This means it includes a term with a variable raised to the second power. Generally, the form of a quadratic polynomial is:

• \(ax^2 + bx + c\)

Here, \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the variable. The coefficient \(a\) cannot be zero because that would make it a linear polynomial, not a quadratic one. In the given exercise, the quadratic polynomial is \(x^2 + 4x + 16\), where the coefficients are \(a = 1\), \(b = 4\), and \(c = 16\). Understanding the structure of a quadratic polynomial is essential for factoring it or determining if it is prime.

discriminant

The discriminant is a critical concept for understanding quadratic polynomials. It is calculated using the formula: \[\text{Discriminant} = b^2 - 4ac\] The discriminant helps determine the nature of the roots of the quadratic equation.

• If the discriminant is positive, there are two distinct real roots.
• If it's zero, there is one real root (a repeated root).
• If the discriminant is negative, there are no real roots; instead, the roots are complex or imaginary.

In our example, we calculated the discriminant as follows: \[4^2 - 4 \times 1 \times 16 = 16 - 64 = -48\] Since the discriminant is \(-48\) (negative), the quadratic polynomial has no real roots. This means it cannot be factored into real-number factors.

prime polynomial

A prime polynomial is a polynomial that cannot be factored further over the set of integers or real numbers. In simpler words, it means there are no two polynomials of lower degrees that can be multiplied together to produce the given polynomial. For our example, \(x^2 + 4x + 16\), we follow these steps:

• First, identify the structure as a quadratic polynomial.
• Second, calculate the discriminant and analyze its value.
• Finally, if the discriminant is negative, as it is in this case, the polynomial is prime over the real numbers.

So, since \(x^2 + 4x + 16\) has a negative discriminant and cannot be factored into real factors, it is a prime polynomial in the context of real numbers.