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Chapter 5: Problem 6

Multiply. $$\left(x^{2}-3\right)(x-1)$$

Short Answer

Expert verified
The product is \( x^3 - x^2 - 3x + 3 \).

Step by step solution

01

Distribute Each Term

Apply the distributive property by multiplying each term in the binomial \(x^{2}-3\) by each term in the binomial \(x-1\). This gives us: \( (x^2)(x) + (x^2)(-1) + (-3)(x) + (-3)(-1) \)
02

Multiply the Terms

Now carry out each multiplication separately: \( (x^2)(x) = x^3 \), \( (x^2)(-1) = -x^2 \), \( (-3)(x) = -3x \), and \( (-3)(-1) = 3 \).
03

Combine Like Terms

Combine all the terms from Step 2 together: \[ x^3 - x^2 - 3x + 3 \]

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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 of algebra. It states that a term outside the parentheses can be multiplied by each term inside the parentheses. In this exercise, we start with the binomials \((x^2-3)\) and \((x-1)\). To use the distributive property, we multiply each term in the first binomial by each term in the second binomial. We get:
  • \((x^2)(x)\)
  • \((x^2)(-1)\)
  • \((-3)(x)\)
  • \((-3)(-1)\)
This step sprawls out the problem, making it more manageable.
Combining Like Terms
Combining like terms is an essential step in simplifying expressions. After applying the distributive property, you often end up with multiple terms that need to be combined. Like terms are terms that have the same variable raised to the same power. Here, after distributing, you'll have:
  • \((x^2)(x) = x^3\)
  • \((x^2)(-1) = -x^2\)
  • \((-3)(x) = -3x\)
  • \((-3)(-1) = 3\)
The next step is to assemble all the terms: \(x^3 - x^2 - 3x + 3\). Since no terms share the same variable and exponent, this is your simplified expression, meaning you can't combine it further.
Binomials
A binomial is a polynomial with exactly two terms. In this example, you are working with two binomials: \((x^2 - 3)\) and \((x - 1)\). Each binomial consists of two terms connected by either addition or subtraction. Binomials are the simplest type of polynomials, often used in algebra to introduce concepts like distribution and combining like terms. Understanding how to multiply binomials is a crucial skill because it forms the basis for more complex algebraic manipulations. The multiplication of these binomials demonstrates the use of both the distributive property and combining like terms.

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