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Chapter 4: Problem 59

Simplify each expression. $$2(x-4)(x+1)$$

Short Answer

Expert verified
The simplified form of the expression is \(2x^2 - 6x - 8\).

Step by step solution

01

Expand the brackets

This is done through the distributive property of multiplication over addition/subtraction. This implies that each term within the bracket should be multiplied by the term outside the bracket. In this case, apply the \(2(x-4)\) to the expression which results in \(2x - 8\). The entire equation now becomes \(2x - 8)(x+1)\).
02

Apply Distributive Property again

At this step, perform the operation of distributive property again over the parenthesis. This results in the multiplication of each of the terms \(2x - 8\) by the terms \(x+1\), which gives \(2x^2 + 2x - 8x - 8\)
03

Combine Like Terms

Combine similar terms together, in this case, the x terms and the constants together. This results in \(2x^2 - 6x - 8\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental principle in algebra which allows us to simplify expressions in a straightforward manner. At its core, it involves multiplying a single term by all terms inside a set of parentheses. This property can be expressed in the form: \(a(b + c) = ab + ac\).
For our original exercise, the expression \(2(x-4)(x+1)\) requires us to apply the distributive property twice.
  • First, distribute the \(2\) across the \((x-4)\), resulting in \(2x - 8\).
  • After this, the expression becomes \((2x - 8)(x+1)\), and we apply the property again to the entire expression.
This step is sometimes referred to as "expanding" because it breaks down and spreads out the expression across multiple terms to simplify further.
Multiplying Expressions
The task of multiplying expressions involves using the distributive property to expand the expressions, ensuring each term from one group is multiplied by every term in the other. This step-by-step multiplication is critical to correctly simplifying polynomials.
In our problem:
  • We take each term in the first expression \((2x - 8)\) and multiply it by each term in the second \((x + 1)\).
  • This results in four separate products: \(2x \cdot x, 2x \cdot 1, -8 \cdot x, \) and \(-8 \cdot 1\).
These calculations yield four terms: \(2x^2, 2x, -8x, \) and \(-8\). Breaking down these multiplications piece by piece helps ensure clarity and accuracy in solving polynomial expressions.
Combining Like Terms
Combining like terms is an essential technique in simplifying polynomial expressions. It involves grouping together terms that share the same variable and exponent. Once terms are expanded using the distributive property, the next step is to simplify the expression further by combining like terms.
In the equation \(2x^2 + 2x - 8x - 8\) from our problem:
  • The expressions \(2x\) and \(-8x\) are like terms because they both have the variable \(x\) raised to the power of 1.
  • By adding these like terms together, \(2x - 8x\) simplifies to \(-6x\).
Thus, our full, simplified expression becomes \(2x^2 - 6x - 8\). Combining like terms is crucial because it condenses the expression into its simplest form, making it easier to understand and use in further calculations.

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