Problem 14 - Elementary Algebra

Question

Factor completely. $$ 50 x 3+25 x_{2}-32 x-16 $$

Step-by-Step Solution

Answer

The expression factors to \((2x + 1)(5x - 4)(5x + 4)\).

Step 1: Group Terms

Group the given polynomial into two pairs for easier factoring. The expression is \( 50x^3 + 25x^2 - 32x - 16 \). We can write it as \( (50x^3 + 25x^2) + (-32x - 16) \).

Step 2: Factor Common Factors in Each Pair

For the first pair \( (50x^3 + 25x^2) \), the common factor is \( 25x^2 \), so we factor to get \( 25x^2(2x + 1) \). For the second pair \( (-32x - 16) \), the common factor is \( -16 \), so we factor to get \( -16(2x + 1) \).

Step 3: Factor out the Common Binomial Factor

Now that both terms have a common binomial factor of \( (2x + 1) \), factor \( (2x + 1) \) out of the expression: \( 25x^2(2x + 1) - 16(2x + 1) = (2x + 1)(25x^2 - 16) \).

Step 4: Factor the Difference of Squares

Notice that \( 25x^2 - 16 \) is a difference of squares, which can be factored as \( (5x)^2 - (4)^2 = (5x - 4)(5x + 4) \). Thus, the expression becomes \( (2x + 1)(5x - 4)(5x + 4) \).

Key Concepts

Common Factor

The concept of a common factor is an integral part of factoring polynomials. A common factor is a number or variable that divides each term in the expression evenly. To begin factoring, you need to identify these common factors within grouped terms.

In our polynomial exercise, when we first group terms into pairs, we spot a common factor in each pair. For example:

• For the pair \(50x^3 + 25x^2\), the common factor is \(25x^2\). Hence, the terms can be expressed as \(25x^2(2x + 1)\).
• For the pair \(-32x - 16\), the greatest common factor is \(-16\), allowing it to be rewritten as \(-16(2x + 1)\).

Recognizing these factors is a key skill, as it is the first step in simplifying polynomials, making them easier to transform using further factoring techniques.

Binomial Factor

A binomial factor is an expression consisting of two terms, commonly seen in polynomial structures. This form is crucial in factoring because, often, both parts of our simplified groups share a binomial factor.

In the example, after factoring out the common factors from each group, both pairs share a binomial factor. Notice that both expressions, \(25x^2(2x + 1)\) and \(-16(2x + 1)\), have \((2x + 1)\) as a factor.

Once identified, this allows us to factor the entire expression by pulling out this common binomial. This means rewriting the polynomial in terms of this shared binomial, enabling further factoring if necessary:

• The expression simplifies to \((2x+1)(25x^2 - 16)\).

Finding common binomial factors simplifies expressions immensely and sets the stage for recognizing more complex structures, like the difference of squares.

Difference of Squares

The difference of squares technique is a powerful tool in polynomial factoring. This method comes into play when a polynomial can be expressed as the difference between two perfect squares.

A perfect square is an expression that can be written as something squared, such as \(a^2\). The difference of squares is characterized by the form \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\).

In the polynomial given, after factoring out the common binomial factor \((2x + 1)\), we're left with another expression: \(25x^2 - 16\). This is a difference of squares because:

• \(25x^2\) is \((5x)^2\)
• \(16\) is \(4^2\)

Thus, it can be factored as \((5x-4)(5x+4)\). Turning differences of squares into products of linear binomials greatly simplifies solving polynomial equations.

Grouping Method

The grouping method is a versatile technique used in polynomial factoring. It is particularly useful when dealing with higher-degree polynomials that have four or more terms.

By grouping terms with common factors, we can break down a seemingly complex polynomial into more manageable pieces.

In our specific problem, we use the grouping method as the first step, organizing \(50x^3 + 25x^2\) and \(-32x - 16\) into separate groups. From there, we factor each group to find common factors:

• The first group, \(50x^3 + 25x^2\), factors as \(25x^2(2x + 1)\)
• The second group, \(-32x - 16\), factors as \(-16(2x + 1)\)

Once common factors in each group are identified, it allows for additional factoring steps, as seen in pulling out a common binomial factor.

This method is foundational in handling polynomials and achieving a fully factored form efficiently.