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Chapter 7: Problem 75

Multiply and simplify each of the following. Whenever possible, do the multiplication of two binomials mentally. $$(x+4)(x-3)+(x-6)(x-2)$$

Short Answer

Expert verified
2x^2 - 7x

Step by step solution

01

- Expand the first binomial product

Expand \( (x+4)(x-3) \) by distributing each term in the first binomial to each term in the second binomial.\[ (x+4)(x-3) = x \cdot x + x \cdot (-3) + 4 \cdot x + 4 \cdot (-3) = x^2 - 3x + 4x - 12 = x^2 + x -12 \]
02

- Expand the second binomial product

Next, expand \( (x-6)(x-2) \) using the same method.\[ (x-6)(x-2) = x \cdot x + x \cdot (-2) - 6 \cdot x - 6 \cdot (-2) = x^2 - 2x - 6x + 12 = x^2 - 8x + 12 \]
03

- Combine the expanded polynomials

Add the expanded results from Step 1 and Step 2.\[ (x^2 + x - 12) + (x^2 - 8x + 12) = x^2 + x - 12 + x^2 - 8x + 12 \]
04

- Simplify the expression

Combine like terms to simplify the resulting polynomial.\[ x^2 + x^2 + x - 8x -12 + 12 = 2x^2 - 7x \]

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Key Concepts

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

Binomial Expansion
Binomial expansion is the process of multiplying two binomials (expressions with two terms). For example, if we have \( (x+4)(x-3) \), we use the distributive property to combine each term in the first binomial with each term in the second binomial. This process is also known as FOIL (First, Outer, Inner, Last).
  • First: Multiply the first terms of each binomial (\( x \cdot x = x^2 \))
  • Outer: Multiply outer terms (\( x \cdot -3 = -3x \))
  • Inner: Multiply inner terms (\( 4 \cdot x = 4x \))
  • Last: Multiply last terms (\( 4 \cdot -3 = -12 \))
Adding these together gives us \( x^2 - 3x + 4x - 12 = x^2 + x - 12 \).We repeat this for the second binomial, \( (x-6)(x-2) \), to get \( x^2 - 2x - 6x + 12 = x^2 - 8x + 12 \). With practice, binomial expansions can be done more mentally by recognizing the patterns.
Polynomial Simplification
Polynomial simplification is the process of making a polynomial expression more compact and easier to understand. After you've expanded the binomials, you often get a polynomial with multiple terms that can be simplified. Simplifying these terms involves combining like terms and performing any necessary operations.Consider the expanded polynomials from the previous step:
  • First expanded polynomial: \( x^2 + x - 12 \)
  • Second expanded polynomial: \( x^2 - 8x + 12 \)
To simplify, we first combine these two polynomials: \( (x^2 + x - 12) + (x^2 - 8x + 12) \). This gives us the combined polynomial: \( x^2 + x^2 + x - 8x - 12 + 12 \).Now, we simplify it by combining the like terms.
Combining Like Terms
Combining like terms is a key step in polynomial simplification, and it ensures that your expression is as simple as possible. Like terms are terms that have the same variables raised to the same power.In our exercise, we identify and combine the like terms as follows:
  • Combine \( x^2 \) terms: \( x^2 + x^2 = 2x^2 \)
  • Combine \(-7x \) terms (since \( x - 8x = -7x \)):
  • Combine \(-12 + 12 \)
So, the simplified polynomial now becomes: \( 2x^2 - 7x \).By combining like terms, we made the polynomial more manageable and prepared it for further operations or evaluations. This skill is essential for solving more complicated algebraic problems efficiently.

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Most popular questions from this chapter

In this section we defined \(a^{0}=1 .\) It is important to realize that a definition is neither right nor wrong-it just is. The proper question to ask about a definition is "Is it useful?" What happens if someone decides on an "alternative definition" such as \(a^{0}=7\) because 7 happens to be his or her favorite number? What happens when we consider the expression \(a^{0} \cdot a^{4} ?\) If you use exponent rule 1 you get \(a^{0} \cdot a^{4}=a^{0+4}=a^{4} .\) However, if you use this alternative definition, you get \(a^{0} \cdot a^{4} \stackrel{\underline{2}}{=} 7 a^{4} .\) The answer we get from the alternative definition is not consistent with the answer we get from the exponent rule. The exponent rules and this alternative definition cannot coexist. Since we do not want to throw away all the exponent rules, we must modify the alternative definition so that it is consistent with all the exponent rules. Verify that our definition of \(a^{0}=1\) is consistent with all five exponent rules.

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you divide \(9.28 \times 10^{7}\) by \(6.86 \times 10^{4},\) the result is closest to (a) 10 (b) 100 (c) 1000 (d) 0.1 (e) 0.01

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you multiply \(2.08 \times 10^{-7}\) by \(5.14 \times 10^{6},\) the result is closest to (a) 10 (b) 100 (c) 1 (d) 0.1 (e) 0.01

Convert each number into standard notation. $$1.76 \times 10^{5}$$

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(3 x+4)(5 x-7)$$
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