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Chapter 11: Problem 56

Multiply and simplify. $$(x-h)^{2}$$

Short Answer

Expert verified
The simplified form is \( x^{2} - 2xh + h^{2} \).

Step by step solution

01

- Expand the Expression

To multiply and simplify \( (x-h)^{2} \), first rewrite it as \( (x-h)(x-h) \).
02

- Apply the Distributive Property

Next, apply the distributive property (also known as the FOIL method for binomials): \( (x-h)(x-h) = x \times x + x \times (-h) + (-h) \times x + (-h) \times (-h) \).
03

- Simplify Each Term

Now, simplify each term: \[ x \times x = x^{2} \] \[ x \times (-h) = -xh \] \[ (-h) \times x = -xh \] \[ (-h) \times (-h) = h^{2} \]
04

- Combine Like Terms

Combine the like terms: \[ (x^{2} - xh - xh + h^{2}) = x^{2} - 2xh + h^{2} \]

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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 key algebraic principle you'll use frequently. It states that multiplying a sum by a number gives the same result as multiplying each addend separately by the number and then adding the products. For example, if you need to expand the expression \((x - h)^{2}\), you can rewrite it as \((x - h)(x - h)\). Now, using the distributive property, you distribute each term in the first binomial across each term in the second binomial. This step is crucial because it helps to break down the problem into smaller, manageable parts, making it easier to solve.
Combining Like Terms
Combining like terms is another essential skill when simplifying algebraic expressions. Like terms are terms that have identical variable parts raised to the same power. For instance, in the expanded form \(x^{2} - xh - xh + h^{2}\), the terms \(-xh\) and \(-xh\) are like terms. To simplify the expression, you combine them by performing the addition or subtraction. In this example:
  • \(x^{2}\) stands alone because there are no other \(x^{2}\) terms.
  • Adding \(-xh -xh\) gives you \(-2xh\).
  • \(h^{2}\) remains as it is since there are no other \(h^{2}\) terms.
Combining these like terms, the simplified expression becomes \(x^{2} - 2xh + h^{2}\). This process makes your solution more concise and easier to interpret.
FOIL Method
The FOIL method is a specific case of the distributive property for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, and it's a step-by-step process for expanding expressions like \((x - h)(x - h)\). Here's how it works: Using FOIL, you multiply each pair of terms in a predictable sequence:
  • First: Multiply the first terms in each binomial (\(x \times x = x^{2}\)).
  • Outer: Multiply the outer terms (\(x \times (-h) = -xh\)).
  • Inner: Multiply the inner terms (\(-h \times x = -xh\)).
  • Last: Multiply the last terms in each binomial (\(-h \times (-h) = h^{2}\)).
After FOIL, your expression looks like: \(x^{2} - xh - xh + h^{2}\). The final step involves combining like terms to simplify, giving you \(x^{2} - 2xh + h^{2}\). Understanding and applying the FOIL method helps you tackle binomial multiplication with confidence.

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Most popular questions from this chapter

In Exercises \(65-74\), solve the given equation. For quadratic equations, choose either the factoring method or the square root method, whichever you think is the easier to use. $$4(x+1)=\frac{9}{x+1}$$

Sketch the graphs of \(y=-x^{2}\) and \(y=-3 x^{2}\) on the same coordinate system. How would you describe the effect the coefficient \(-3\) has on the graph of \(y=x^{2} ?\)

Suppose that a company's profit \(P,\) in thousands of dollars, on the sale of \(x\) items is given by the equation. $$P=x^{2}-6 x-40 \quad \text { where } x \geq 0$$ (a) Sketch the graph of this equation. Let \(x\) be the horizontal axis and \(P\) be the vertical axis. (b) Use the graph to determine the minimum profit the company earns. (c) How many items does the company sell if it experiences this minimum profit?

Solve each of the following exercises algebraically. The width of a rectangle is one third its length. If the area of the rectangle is \(20 \mathrm{sq}\) in., what are the dimensions of the rectangle?

Sketch the graph of the given equation. Find the intercepts; approximate to the nearest tenth where necessary. $$y=6 x-2 x^{2}$$
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