Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 2-8 x^{2} $$

Short answer

2(1 + 2x)(1 - 2x)

Step-by-step solution

- Identify common factors

First, notice if there are any common factors in the polynomial. In the polynomial \(2-8x^2\), the number 2 is a common factor.

- Factor out the common factor

Extract the common factor from each term in the polynomial. Factor out 2 from both terms: \[2 - 8x^2 = 2(1 - 4x^2)\]

- Recognize the difference of squares

Observe that the expression inside the parentheses, \(1 - 4x^2\), is a difference of squares, because \(1 = (1)^2\) and \(4x^2 = (2x)^2\). The formula for factoring a difference of squares is \(a^2 - b^2 = (a + b)(a - b)\).

- Apply the difference of squares formula

Factor using the difference of squares formula where \(a = 1\) and \(b = 2x\):\[1 - 4x^2 = (1 + 2x)(1 - 2x)\]

- Write the completely factored form

Combine the common factor extracted in Step 2 with the factored difference of squares:\[2(1 - 4x^2) = 2(1 + 2x)(1 - 2x)\]

Key concepts

Common Factors

To start factoring any polynomial, always look for common factors. A common factor is a number or variable that evenly divides each term of the polynomial. In our given exercise, the polynomial is \(2 - 8x^2\). Here, both terms 2 and \(-8x^2\) share the common factor of 2. Identifying this simplifies the problem significantly, as you can pull out this common factor to make the polynomial easier to work with.
Once you recognize the common factor, factor it out from each term. For \(2 - 8x^2\), you get \(2(1 - 4x^2)\). Factoring out the common factor is crucial because it often reveals other factoring opportunities within the polynomial.

Difference of Squares

A difference of squares is a special type of polynomial given by the form \(a^2 - b^2\). This pattern can be factored to \((a + b)(a - b)\). It's essential to identify it because it simplifies otherwise complex polynomial expressions.
In our exercise, after factoring out the common factor, we're left with \(1 - 4x^2\). Notice that both 1 and \(4x^2\) are perfect squares, as \(1 = (1)^2\) and \(4x^2 = (2x)^2\). This matches the difference of squares form \(a^2 - b^2\). Applying the difference of squares here, we get \((1 + 2x)(1 - 2x)\). Recognizing and applying the difference of squares property is key to fully breaking down and factoring certain polynomials.

Completely Factored Form

Finding the completely factored form means breaking the polynomial down into its simplest components, where none of the factors can be factored further. Using our exercise as an example, we first identified and factored out the common factor to get \(2(1 - 4x^2)\).
Next, we factored the inside expression using the difference of squares to get \(2(1 + 2x)(1 - 2x)\). Check each part to ensure they cannot be factored further. The completely factored form of the polynomial \(2 - 8x^2\) is \(2(1 + 2x)(1 - 2x)\). This decomposition makes solving equations or simplifying expressions involving the polynomial much more straightforward.