Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 5+16 x-16 x^{2} $$

Short answer

The factored form is \((4x - 5)(-4x - 1)\).

Step-by-step solution

- Identify the polynomial format

The given polynomial is: \(5 + 16x - 16x^2\). Let us rewrite it in standard form: \(-16x^2 + 16x + 5\).

- Check for a common factor

Look for any common factors in the coefficients of \(-16x^2 + 16x + 5\). There are no common factors among -16, 16, and 5.

- Factor by grouping and trial

First, apply factoring to get the polynomial in a more factorable form. We look for two numbers that multiply to \(a \times c = -80\) (from \(-16 \times 5\)) and add up to \(b = 16\). These numbers are 20 and -4. Rewrite the polynomial as: \(-16x^2 + 20x - 4x + 5\).

- Grouping the middle terms

Group the terms: \[(-16x^2 + 20x) + (-4x + 5)\].

- Factor out the greatest common factor in each group

Factor out the common factors in each group: \(-4x(4x - 5) - 1(4x - 5)\).

- Recognize the common binomial factor

Notice the binomial \((4x - 5)\) is common in both groups, so factor it out: \((4x - 5)(-4x - 1)\).

Key concepts

Polynomial Functions

A polynomial function is an expression composed of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. An important feature of polynomial functions is that their exponents are non-negative integers. For example, the polynomial function in our exercise is given in the form of \texpression: \(5 + 16x - 16x^2\).
Polynomials can be classified based on their degree:

• A linear polynomial (degree 1)
• A quadratic polynomial (degree 2)
• A cubic polynomial (degree 3)

In our exercise, we have a quadratic polynomial since the highest exponent of x is 2. Understanding how to manipulate and factor these functions is crucial for solving various algebraic problems.

Factoring Techniques

Factoring involves writing a polynomial as a product of its factors. There are several techniques for factoring polynomials, including:

• Factoring by grouping
• Using the distributive property
• Applying special formulas like difference of squares

In the given exercise, we use the 'factoring by grouping' method. This method involves arranging the polynomial in such a way that pairs of terms can be grouped together, then factoring out common factors from each group. Let’s go through a step from the solution:
We rewrite \(-16x^2 + 16x + 5\) in a form that can be grouped: \(-16x^2 + 20x - 4x + 5\).
Next, group the terms: \[ (-16x^2 + 20x) + (-4x + 5) \]. This method simplifies the polynomial into manageable parts for further factoring.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can represent real-world scenarios or theoretical problems. The goal in algebra is often to simplify these expressions or solve equations involving them.
The given polynomial \(5 + 16x - 16x^2\) can be seen as an algebraic expression. To factor it, noticing patterns and common parts within the expression is key.
For instance, recognizing \(20x\) and \(-4x\) as terms helps us to group and simplify the polynomial: \[ -4x(4x - 5) - 1(4x - 5) \].
The important skill is to manipulate algebraic expressions to find simpler or equivalent forms, which makes solving algebraic equations more straightforward.

Binomial Factors

A binomial is an algebraic expression containing exactly two terms. When factoring polynomials, we try to rewrite them as products of simpler binomials. In this exercise, one of the binomial factors we obtain is \(4x - 5\).
To get there, we factor out common factors from grouped terms: \[ -4x(4x - 5) - 1(4x - 5) \].
Notice that \(4x - 5\) appears in both groups, which is a common binomial factor. The complete factorization of our polynomial then becomes: \( (4x - 5)(-4x - 1) \).
The ability to identify and factor out binomial terms is crucial, as it simplifies the polynomial into a product of binomials, making it easier to solve or further simplify.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor shared by all terms in a polynomial. Identifying the GCF is an important step in factoring, as it simplifies the process.
In our polynomial \(-16x^2 + 16x + 5\), we first check if there is a common factor among all coefficients and terms. Here, -16, 16, and 5 have no shared common factor, meaning the GCF for these terms is 1.
When working with groups of terms, we can often find common factors within each group. For example, in the terms \(-16x^2 + 20x\), the common factor is \(-4x\), giving us: \(-4x(4x - 5)\).
Identifying and factoring out the GCF simplifies polynomials and paves the way for complete factorization.