Problem 1 - Algebra for College Students

Question

After reading this section, write out the answers to these questions. Use complete sentences. How can you use the discriminant to determine if a quadratic polynomial can be factored?

Step-by-Step Solution

Answer

A quadratic polynomial can be factored if its discriminant is a perfect square.

Step 1: Understand the Discriminant

The discriminant of a quadratic polynomial in the form y = ax^2 + bx + c is given by the formula D = b^2 - 4ac. It helps to determine the nature of the roots of the quadratic equation.

Step 2: Discriminant and Factoring

If the discriminant (D) is a perfect square (like 0, 1, 4, 9, 16, etc.), then the quadratic polynomial can be factored into rational numbers. Otherwise, it cannot be factored over the set of rational numbers.

Step 3: Applying the Concept

To check if a specific quadratic polynomial can be factored, calculate the discriminant D, and then check if D is a perfect square. For example, for the polynomial 2x^2 - 4x - 6, D = (-4)^2 - 4(2)(-6) = 16 + 48 = 64 which is a perfect square, indicating that it can be factored.

Step 4: Conclusion

Summarize the results: The discriminant helps to determine if a quadratic polynomial can be factored by checking whether it is a perfect square. If it is a perfect square, the polynomial can be factored.

Key Concepts

Factoring Quadratic Polynomials

Factoring quadratic polynomials is an essential skill in algebra. A quadratic polynomial is generally of the form \(ax^2 + bx + c\). Factoring involves expressing this polynomial as a product of two binomials. For example, \(x^2 + 5x + 6\) can be factored into \((x + 2)(x + 3)\). This process requires us to find two numbers that multiply to give the constant term (c) and add to give the coefficient of the linear term (b).
Understanding how to factor these expressions is vital for solving quadratic equations and simplifying expressions in algebra.

Nature of Roots

The nature of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is determined by the discriminant \(D\), which is found using the formula \(D = b^2 - 4ac\). The value of the discriminant tells us about the roots or solutions of the quadratic equation:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root (also known as a repeated or double root).
• If \(D < 0\), there are no real roots, but two complex roots.

Knowing this helps to ascertain whether the quadratic equation has real solutions, and if so, how many.

Perfect Squares in Algebra

In algebra, a perfect square is a polynomial that can be expressed as the square of a binomial. For example, \(x^2 + 6x + 9\) is a perfect square because it is equivalent to \((x + 3)^2\).
Identifying perfect squares is useful in factoring and solving quadratic equations.
Additionally, when the discriminant (\(D = b^2 - 4ac\)) of a quadratic polynomial is a perfect square, it suggests that the polynomial can be factored into rational roots. This knowledge significantly simplifies the process of solving quadratic equations, as it indicates that the roots are rational and gives a clear method for factoring the polynomial.