Problem 1 - College Algebra Essentials

Question

By factoring and then using the zero-product principle. $$ 3 x^{4}-48 x^{2}=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(3x^{4}-48x^{2}=0\) are \(x=0\), \(x=4\), and \(x=-4\).

Step 1: Factor out the greatest common factor

Notice that the given expression, \(3x^{4}-48x^{2}\), has a common factor of \(3x^{2}\). Factoring this out gives \(3x^2(x^2 - 16)\).

Step 2: Factorize the quadratic

The term within the brackets, \(x^2 - 16\), is a simple difference of two squares, which can be factored into \((x-4)(x+4)\). So the equation becomes \(3x^2(x-4)(x+4) = 0\).

Step 3: Apply the Zero Product Property

Now that the equation is a product of factors equalling zero, apply the zero product property, which states if ab=0, then a=0 or b=0. Set each factor equal to zero and solve for x. From this, we obtain \(x=0\), \(x=4\), \(x=-4\).

Key Concepts

Factoring quadratic equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). These can often be factored into the product of two binomials. Factoring helps in solving these equations, especially when they are set to zero, as it allows you to apply the Zero Product Property.
A quadratic equation can be factored by following a few steps:

• Identify a common factor.
• Check if the quadratic is a perfect square or if it can be rearranged as the product of two binomials.
• Use methods like splitting the middle term, or applying formulas for special products, such as the difference of squares.

Once the quadratic is factored, each factor can be set to zero to find the solutions. This is because of the Zero Product Property, which concludes that if one factor is zero in a product, the entire product is zero.

Difference of squares

In algebra, certain types of quadratics can be simplified using special factorizations. One of these is the "difference of squares." It is identified as an expression of the form \(a^2 - b^2\). This can be factored into \((a - b)(a + b)\).
To adapt this to an algebra question:

• Look for terms that can be written as squares.
• Identify the subtraction sign between them, suggesting the 'difference'.
• Factor using the formula, knowing both terms should have no middle component, just like \((a - b)(a + b)\).

Using this technique simplifies the equation, making it easier to apply the Zero Product Property and find solutions.

Greatest common factor

One crucial strategy in factoring is to determine the greatest common factor (GCF). The GCF is the highest number or expression that divides all the terms in a polynomial without leaving a remainder.
To find the GCF:

• Examine the coefficients of each term and find the largest integer that divides them.
• Check for shared variables, and take the smallest power of these variables common to all terms.

Factoring out the GCF simplifies the polynomial, reducing it to an easier-to-manage expression, potentially revealing further factorization opportunities.
In essence, factoring out the GCF is like pulling out a common element from a group, setting the stage for applying other factorization strategies.