Problem 12 - Beginning and Intermediate Algebra

Question

Factor by grouping. Find the polynomial which factors to \((3 x-8)(x+2)\)

Step-by-Step Solution

Answer

The polynomial which factors to \((3x-8)(x+2)\) is \(3x^2 - 2x - 16\).

Step 1: Expand the expression using the distributive property

In this step, we will multiply each term of the first expression by each term of the second expression using the distributive property. \((3x-8)(x+2) = 3x \cdot (x+2) - 8 \cdot (x+2)\)

Step 2: Multiply each term in the expressions individually

Now, we will multiply each term in the expressions: \(3x(x+2) = 3x^2 + 6x\) \(-8(x+2) = -8x -16\)

Step 3: Combine the results of the multiplication

Now that we have the products of each term, we can combine them to get the polynomial: \(3x^2 + 6x - 8x - 16\)

Step 4: Simplify the polynomial

Combine like terms to get the simplified polynomial: \(3x^2 - 2x - 16\) So, the polynomial which factors to \((3x-8)(x+2)\) is \(3x^2 - 2x - 16\).

Key Concepts

Distributive Property

The distributive property is like a shopping at a market where you wish to distribute your purchases equally. This property allows us to expand an expression by distributing each term of one factor to each term in another factor. It's why we call it a property of distribution.
In math, to use the distributive property, we multiply each term in one set of parentheses by every term in another set of parentheses. This is expressed mathematically as:

• If you have an expression: \((a+b)(c+d)\), the expansion involves: \(a(c+d) + b(c+d)\).

In this way, every term in the first group is combined with every term in the second group. It’s an essential property that lays the foundation for polynomial expansion, as seen in the given step-by-step solution.
In our example, we distribute:

• \(3x\) across \((x+2)\)
• \(-8\) across \((x+2)\).

This method enables the transformation of the expression \((3x-8)(x+2)\) into expanded terms that can be further simplified.

Polynomial Expansion

Polynomial expansion is like opening up a packed suitcase. Each item inside needs to be laid out so you can see everything clearly. Here, once we apply the distributive property, we multiply terms and express the result as a single polynomial.
The transformation from a product to a sum or difference of terms is called polynomial expansion. It involves taking the originally compact form of an expression and writing it as a series of separate terms.

• For instance, from the original product \((3x-8)(x+2)\), the expansion walks us through individual multiplications like \(3x \cdot x, 3x \cdot 2, -8 \cdot x\), and \(-8 \cdot 2\).

Each result from this multiplication becomes its own term in a polynomial. This is how we get \(3x^2 + 6x - 8x - 16\) from our example. These new terms combine to form an expanded version of the earlier product, expressed as a polynomial, clearly showing each part.

Combining Like Terms

After expanding a polynomial, the equation may look "cluttered" because of similar terms. Combining like terms is akin to grouping similar items together.
When we say "like terms," we refer to terms that have the same variables raised to the same power. Only coefficients can differ. This simplification step keeps our mathematical work neat and concise.

• For example, in the expression \(3x^2 + 6x - 8x - 16\), the terms \(6x\) and \(-8x\) are like terms with the variable \(x\).
• Similarly, constant terms or plain numbers are also considered like terms in expressions such as \(3 + 4\).

By combining \(6x\) and \(-8x\), we simplify this polynomial to \(3x^2 - 2x - 16\). It's more concise and easier to understand. This simplification is crucial in reducing errors and can make further operations clearer and simpler.