Problem 38 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 3 w^{2}+5 w-7 $$

Step-by-Step Solution

Answer

The polynomial \(3w^2 + 5w - 7\) is a prime polynomial.

Step 1: Identify the coefficients

The given polynomial is in the standard form of a quadratic equation: \[3w^2 + 5w - 7\]. Identify the coefficients: \[a = 3, b = 5, c = -7\].

Step 2: Calculate the discriminant

Use the formula for the discriminant of a quadratic equation: \[D = b^2 - 4ac\]. Substitute the values of \(a\), \(b\), and \(c\): \[D = 5^2 - 4 \times 3 \times (-7) = 25 + 84 = 109\].

Step 3: Determine the nature of the discriminant

Check the value of the discriminant: \[D = 109\]. Since the discriminant is not a perfect square, the quadratic expression cannot be factored using rational numbers.

Step 4: Conclusion

Since the discriminant is not a perfect square, the polynomial \[3w^2 + 5w - 7\] cannot be factored into simpler polynomials with rational coefficients. Therefore, it is a prime polynomial.

Key Concepts

Quadratic Equations

Quadratic equations are essential in algebra and appear in the form ax^2 + bx + c = 0, where a, b, and c are coefficients. Here, 'a' cannot be zero.
This equation represents a parabola when graphed on a coordinate plane. Quadratic equations can be solved using various methods, including:

• Factoring
• Quadratic Formula
• Completing the Square

In the case of the given polynomial 3w^2 + 5w - 7, we determine whether it can be factored by evaluating its discriminant.

Discriminant

The discriminant ( D ) helps determine the nature of the roots of a quadratic equation. Given a quadratic equation of the form ax^2 + bx + c, the discriminant is calculated as: D = b^2 - 4ac. Based on the value of the discriminant, we can understand the roots:

• If D > 0 and a perfect square, the quadratic can be factored into real rational numbers.
• If D > 0 but not a perfect square, the roots are real but irrational.
• If D = 0, the quadratic has a single (repeated) rational root.
• If D < 0, the quadratic has complex roots (not real).

In the exercise, we found D = 109, which is not a perfect square. Hence, the polynomial 3w^2 + 5w - 7 cannot be factored over the rational numbers.

Prime Polynomials

A polynomial is considered prime if it cannot be factored into the product of polynomials with smaller degrees. Identifying prime polynomials involves checking if they can be broken down using any factoring methods.
In our exercise, since the discriminant ( D = 109 ) is not a perfect square, it indicates no real, rational factors exist for the polynomial 3w^2 + 5w - 7. Thus, it remains in its simplest form and is considered a prime polynomial.
Knowing how to identify and work with prime polynomials can significantly simplify solving algebraic problems. This knowledge helps determine when to stop attempting to factor and recognize the polynomial's inherent prime nature.