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Chapter 5: Problem 47

Find each product. \(x^{2}(x-2)\)

Short Answer

Expert verified
\(x^3 - 2x^2\)

Step by step solution

01

- Distribute the term

Distribute the term outside the parenthesis, which is \(x^2\), to each term inside the parenthesis. Specifically, multiply \(x^2\) by \(x\) and then by \(-2\).
02

- Multiply the first term

First, multiply \(x^2\) by \(x\). When multiplying exponents with the same base, add the exponents: \(x^2 \cdot x = x^{2+1} = x^3\).
03

- Multiply the second term

Next, multiply \(x^2\) by \(-2\). Since \(-2\) is a constant, simply multiply: \(x^2 \cdot (-2) = -2x^2\).
04

- Combine the results

Combine the results from the previous steps: \(x^3 - 2x^2\).

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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 concept in algebra. It allows us to simplify expressions and equations. When you see an expression like \(a(b + c)\), the distributive property tells us that it can be expanded to \(ab + ac\). This principle applies to any mathematical operation.
In our exercise, we used it to distribute \(x^2\) to each term inside the parenthesis \((x-2)\). We did it in two steps: multiplying \(x^2\) by \(x\) and by \(-2\).
This simplifies the multiplication and makes it easier to work with polynomial expressions.
Exponent Rules
Exponent rules help us manage and simplify expressions involving powers of the same base. One important rule is the product of powers rule. It states that when you multiply two exponents with the same base, you add their exponents. For example, \(x^a \cdot x^b = x^{a+b}\).
In the exercise, when we multiplied \(x^2\) by \(x\), we added their exponents: \[x^2 \cdot x = x^{2+1} = x^3\]. This rule simplifies expressions and makes polynomial multiplication more manageable.
Combining Like Terms
Combining like terms simplifies polynomial expressions. Like terms are terms with the same variable raised to the same power. For example, \(3x^2\) and \(-2x^2\) are like terms, but \(3x^2\) and \(-2x\) are not.
In our exercise, if we had other terms with \(x^3\) or \(x^2\), we would combine them to simplify the expression. This ensures that our final expression is as simple as possible. It makes it easier to understand and work with polynomial equations.
Always remember to look out for like terms and combine them whenever possible to keep your expressions neat and simple.

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Most popular questions from this chapter

Solve each equation. $$(2 x-1)(3 x+1)(4 x-1)=0$$

If you divide each side of \(x^{2}=x\) by \(x,\) you get \(x=1\) If you subtract \(x\) from each side of \(x^{2}=x,\) you get \(x^{2}-x=0,\) which has two solutions. Which method is correct? Explain.

Factor each polynomial by grouping. $$ 2 q h-h+8 q-4 $$

Factor each polynomial using the trial-and-error method. $$ a^{2}-9 a+18 $$

Factor each polynomial by grouping. $$ y^{2}-a y-b y+a b $$
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