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Chapter 5: Problem 56

Find each product. In each case, neither factor is a monomial. $$(x+4)(x+6)$$

Short Answer

Expert verified
The product of \((x+4)(x+6)\) is \(x^2 + 10x + 24\).

Step by step solution

01

Apply the FOIL method

Distribute each term in the first binomial over the second binomial, which means first multiply the first terms, multiply outers, multiply inners, and multiply lasts. Hence: \[(x+4)(x+6) = x*x + x*6 + 4*x + 4*6\]
02

Simplify multiplication expressions

Simplify each multiplication from the first step, which results in: \[x^2 + 6x + 4x + 24\]
03

Combine like-terms

Combine terms that have the same variable to the same power, which gives us: \[x^2 + 10x + 24\]

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Key Concepts

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

FOIL Method
The FOIL Method is an important technique used in algebra for multiplying two binomials. It's an abbreviation that stands for First, Outer, Inner, Last. This refers to the order in which you multiply the terms from each binomial. When you are given a problem such as \((x+4)(x+6)\), you can apply the FOIL method to find the product as follows:
  • First: Multiply the first term in each binomial: \(x \cdot x = x^2\).
  • Outer: Multiply the outer terms in the expression: \(x \cdot 6 = 6x\).
  • Inner: Multiply the inner terms: \(4 \cdot x = 4x\).
  • Last: Multiply the last terms in each binomial: \(4 \cdot 6 = 24\).
Putting it all together, you have the sum \(x^2 + 6x + 4x + 24\). Using the FOIL method helps ensure no part of the binomials is overlooked and makes the process methodical and straightforward.
Combining Like Terms
Combining Like Terms is a crucial part of simplifying polynomial expressions. Once you've used the FOIL method to multiply binomials, you'll often end up with terms that can be combined. These are terms that have the same base and power. After applying FOIL to \((x+4)(x+6)\), you'll have the expression \(x^2 + 6x + 4x + 24\).Here鈥檚 how you combine like terms:
  • The terms \(6x\) and \(4x\) are like terms because they both have the variable \(x\) raised to the first power.
  • Add these coefficients together: \(6 + 4 = 10\). So, the terms combine to form \(10x\).
After combining like terms, the expression becomes \(x^2 + 10x + 24\). This is a simpler and cleaner form of the polynomial.
Polynomial Expressions
Understanding Polynomial Expressions is key in algebra. A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents. For example, \(x^2 + 10x + 24\) is a polynomial expression.Polynomials can have one or more terms:
  • The term \(x^2\) is a "quadratic term" because the variable \(x\) is squared.
  • The term \(10x\) is a "linear term" because \(x\) has an exponent of 1.
  • The number \(24\) is a "constant term" as it doesn't have a variable attached to it.
Polynomials are widely used in various areas of mathematics and real-world applications. Recognizing the structure of polynomial expressions helps in performing operations such as addition, subtraction, and multiplication of polynomials effectively. In our case, converting the binomials into a single polynomial through multiplication and simplification is part of understanding these expressions.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I divide monomials by dividing coefficients and subtracting exponents.

Solve each system by the method of your choice. $$\left\\{\begin{array}{l}3 x+4 y=7 \\\2 x+7 y=9\end{array}\right.$$

What is a monomial? Give an example with your explanation.

In Exercises \(156-163\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$

The mad Dr. Frankenstein has gathered enough bits and pieces (so to speak) for \(2^{-1}+2^{-2}\) of his creature-to-be. Write a fraction that represents the amount of his creature that must still be obtained.
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