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Chapter 4: Problem 52

Find each product. $$ (n+9)(n+3) $$

Short Answer

Expert verified
(n)(n) + (9)(n) + (n)(3) + (9)(3) leading to simplified form of FINAL ANSWER PLACE HOLDER.

Step by step solution

01

Distribute the terms

Use the distributive property (also known as the FOIL method for binomials) to expand the product Multiply each term in the first binomial by each term in the second binomial: ... Divide the terms as follows: ... and organize the products into the expanded form.

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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 algebraic principle used to simplify expressions and solve equations. It states that for any numbers or variables a, b, and c:
\( a(b + c) = ab + ac\)
For our exercise, we apply the distributive property to the binomials \( (n + 9)(n + 3) \). This means we'll distribute (or multiply) each term in the first binomial by each term in the second binomial.

\( n \times n \)
\( n \times 3 \)
\( 9 \times n \)
\( 9 \times 3 \)
By doing this, we get an expression where all terms have been individually multiplied, helping us to expand and ultimately simplify the product.
FOIL Method
The FOIL method is a special case of the distributive property, specifically for multiplying two binomials. FOIL stands for First, Outer, Inner, Last.
  • First: Multiply the first terms in each binomial: \( n \times n \)
  • Outer: Multiply the outer terms in the binomials: \( n \times 3 \)
  • Inner: Multiply the inner terms in the binomials: \( 9 \times n \)
  • Last: Multiply the last terms in each binomial: \( 9 \times 3 \)
Applying the FOIL method, we take each term and multiply it accordingly:
\( (n + 9)(n + 3) = n^2 + 3n + 9n + 27 \).
Each product corresponds to one of the FOIL steps, making it an easy way to remember how to multiply two binomials. Finally, we combine like terms to get our expanded expression.
Expanding Products
Expanding products is the process of multiplying and simplifying the terms of an algebraic expression. For the binomials \( (n + 9)(n + 3) \), expanding involves several steps:
  • Distribute each term in one binomial across every term in the second binomial.
  • Use the FOIL method for a structured approach when dealing with binomials.
  • Combine like terms to simplify the expression.
Starting with:
\( n^2 + 3n + 9n + 27 \),
we look for like terms to combine. Here, \( 3n \) and \( 9n \) are like terms:
\( 3n + 9n = 12n \).
Thus, the fully expanded and simplified product is:
\( n^2 + 12n + 27 \).
This expanded form is more straightforward to work with and allows for easier interpretation and further algebraic manipulations.

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Most popular questions from this chapter

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 20 \frac{1}{2} \times 19 \frac{1}{2} $$

$$ \begin{aligned} &y=-x^{2}+2\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

Add or subtract as indicated. \((6 b+3 c)+(-2 b-8 c)\)

The distance to Earth from Pluto is \(4.58 \times 10^{9} \mathrm{~km}\). Pioneer 10 transmitted radio signals from Pluto to Earth at the speed of light, \(3.00 \times 10^{5} \mathrm{~km}\) per sec. About how long (in seconds) did it take for the signals to reach Earth?

Find each product. $$ -5 t^{4}\left(2 t^{4}+1\right)\left(2 t^{4}-1\right) $$
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