Problem 70 - Elementary Algebra

Question

The volume of a certain right circular cylinder is given by the function \(V(x)=4 \pi x 3-4 \pi x 2+\pi x .\) Write the function in its factored form.

Step-by-Step Solution

Answer

The factored form is \( V(x) = \pi x (2x - 1)^2 \).

Step 1: Identify Common Terms

Look for the greatest common factor in all the terms of the polynomial. The function given is: \[ V(x) = 4\pi x^3 - 4\pi x^2 + \pi x \]The term \(\pi x\) is common in all terms.

Step 2: Factor Out the Greatest Common Factor

Factor out \( \pi x \) from the expression:\[ V(x) = \pi x (4x^2 - 4x + 1) \]

Step 3: Check for Further Factoring

Examine the quadratic part \(4x^2 - 4x + 1\) to see if it can be factored further. Here, apply the quadratic factoring formula or check if it factors into perfect squares.

Step 4: Use the Quadratic Formula (if necessary)

To see if the expression can be factored further, use the quadratic formula. For \(4x^2 - 4x + 1\), the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) gives us:Where \(a = 4\), \(b = -4\), \(c = 1\):\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} \]This simplifies to \(x = \frac{4 \pm 0}{8} = \frac{1}{2}\).

Step 5: Conclude on Factoring

Since the discriminant \((b^2 - 4ac) = 0\), the quadratic \(4x^2 - 4x + 1\) is a perfect square, which implies:\[ 4x^2 - 4x + 1 = (2x - 1)^2 \]Thus, the completely factored form of the volume function is:\[ V(x) = \pi x (2x - 1)^2 \]

Key Concepts

Greatest Common Factor

When working with polynomials, the first step in factoring is often to find the Greatest Common Factor (GCF). This is the largest expression that divides each of the terms in the polynomial without leaving a remainder. Finding the GCF simplifies the factoring process by reducing the expression to a more manageable form.
For the function given, \( V(x) = 4\pi x^3 - 4\pi x^2 + \pi x \), we notice that each term includes the factor \( \pi x \). Thus, \( \pi x \) is the GCF. By factoring \( \pi x \) out of the polynomial, we get:

• \( \pi x (4x^2 - 4x + 1) \)

This initial step makes further factoring a lot simpler, as it focuses on the remaining quadratic expression.

Quadratic Formula

The quadratic formula is a reliable method for solving quadratic equations, particularly when they cannot be easily factored using other methods. The formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

It allows you to find the roots or solutions of any quadratic equation of the form \( ax^2 + bx + c = 0 \).
In the exercise, after factoring out the GCF, we are left with the quadratic expression \( 4x^2 - 4x + 1 \). Here, using the quadratic formula helps us determine the characteristics of this quadratic expression and whether it can be further simplified or expressed in a special form.

Discriminant

A key part of the quadratic formula is the discriminant, which is the expression under the square root: \( b^2 - 4ac \). The discriminant tells us about the nature of the roots of the quadratic equation:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is one real root, or a repeated root.
• If it is negative, there are no real roots, only complex ones.

In our specific problem, the discriminant for \( 4x^2 - 4x + 1 \) is:

• \[ (-4)^2 - 4 \times 4 \times 1 = 16 - 16 = 0 \]

Since the discriminant is zero, it means our quadratic is a perfect square, indicating it can be rewritten as the square of a binomial.

Perfect Square

A perfect square in the context of quadratics is an expression that can be rewritten as the square of a binomial.
When we have a quadratic expression like \( 4x^2 - 4x + 1 \), and the discriminant is zero, it can be expressed as a perfect square. This means that the expression takes the form of \( (px + q)^2 \).
From the exercise, we reach the conclusion that \( 4x^2 - 4x + 1 \) simplifies to \( (2x - 1)^2 \). This form confirms that the quadratic is indeed a perfect square.
Thus, the completely factored form of the original polynomial is \( V(x) = \pi x (2x - 1)^2 \). Factoring in this manner showcases the symmetry and simplicity sometimes hidden within quadratic expressions, and highlights the elegance of perfect squares in mathematical structures.