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Chapter 2: Problem 7

Expand and combine like terms. $$ 3(x-4)^{2}+8 x-48 $$

Short Answer

Expert verified
Answer: The simplified form of the expression is \(3x^2 - 16x\).

Step by step solution

01

Expand the squared term

First, let's expand the term \((x-4)^2\). To do this, we will use the binomial formula: \((a-b)^2 = a^2 - 2ab + b^2\). In our case, \(a=x\) and \(b=4\). Expanding \((x-4)^2\), we get: $$ (x-4)^2 = x^2 - 2(x)(4) + 4^2 $$
02

Substitute the expanded term

Now, let's replace \((x-4)^2\) in the given expression with the expanded form that we found in the previous step: $$ 3(x^2 - 8x + 16) + 8x - 48 $$
03

Distribute the constants

Now, we need to distribute the constants to the terms inside the parentheses: $$ 3x^2 - 24x + 48 + 8x - 48 $$
04

Combine like terms

Finally, let's combine the like terms in the expression: $$ 3x^2 - 24x + 8x + 48 - 48 $$ The simplified expression is: $$ 3x^2 - 16x $$

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Key Concepts

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

Binomial Expansion
When dealing with binomial expressions, expanding them can be a valuable skill. Binomial expressions typically look like \((a + b)^n\) or \((a - b)^n\), and when expanded, they show the influence of both terms raised to a power. In our case, \((x-4)^2\) was the expression to expand. Using the formula \((a-b)^2 = a^2 - 2ab + b^2\), we were able to expand it into three separate terms: \(x^2 - 2 \times x \times 4 + 4^2\).
  • This method can simplify how we tackle polynomial terms by breaking them down into manageable parts.
  • Remember that each component, such as \(-2ab\), arises from the interactions between elements \(a\) and \(b\).
Mastering binomial expansion makes it easier to simplify polynomials and solve algebraic equations more efficiently.
Like Terms
Combining like terms is crucial for simplifying an expression. "Like terms" are terms in an expression that have identical variable parts, like \(x^2\) or \(x\). For example, in the expression \(3x^2 - 24x + 8x + 48 - 48\), the terms \(-24x\) and \(8x\) are like terms because they share the same variable \(x\).
  • To combine these, we simply add or subtract their coefficients, resulting in \(-16x\).
  • Constant numbers, like \(48\) and \(-48\), are also combined in the same way.
This process helps to simplify the equation to its most reduced form, making it much easier to solve or interpret. By focusing on like terms, we can zero in on the significant parts of the expression.
Distributive Property
The distributive property is a valuable tool when working with algebraic expressions. It allows us to multiply a single term by a sum or difference within parentheses, essentially distributing the multiplication across each term inside. In our exercise, the equation \(3(x^2 - 8x + 16)\) uses this property effectively:
  • Multiply \(3\) across each term inside the parentheses: \(3 \times x^2\), \(3 \times -8x\), \(3 \times 16\).
  • This results in the expanded version: \(3x^2 - 24x + 48\).
Utilizing this property ensures more complex expressions can be broken down into simpler, manageable parts, facilitating easier computation and further simplification. It's a fundamental algebraic technique that appears in many areas of mathematics.

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Most popular questions from this chapter

Rewrite the expression by taking out the common factors. \(-4 a^{2} b-6 a b^{2}-2 a b\)

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A contractor is managing three different job sites. It costs her $$\$ c$$ to employ a carpenter, $$\$ p$$ to employ a plumber, and $$\$ e$$ to employ an electrician. The total cost to employ carpenters, plumbers, and electricians at each site is Cost at site \(1=12 c+2 p+4 e\) Cost at site \(2=14 c+5 p+3 e\) Cost at site \(3=17 c+p+5 e\). Write expressions in terms of \(c, p,\) and \(e\) for: (a) The total employment cost for all three sites. (b) The difference between the employment cost at site 1 and site 3 . (c) The amount remaining in the contractor's budget after accounting for the employment cost at all three sites, given that originally the budget is \(50 c+10 p+20 e\)

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