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Chapter 4: Problem 4

Find each product. $$ (x+8)^{2} $$

Short Answer

Expert verified
\( x^2 + 16x + 64 \)

Step by step solution

01

Recognize the Formula

This problem requires expanding the square of a binomial. We will use the formula to expand \( (a+b)^2 = a^2 + 2ab + b^2 \).
02

Identify Variables

In this case, \( (x + 8)^2 \) matches the format \( (a + b)^2 \). Here, \( a = x \) and \( b = 8 \).
03

Apply the Formula

Substitute \( a \) and \( b \) into the formula: the formula: \( (a+b)^2 = a^2 + 2ab + b^2 \)then \( (x+8)^2 = x^2 + 2(x)(8) + 8^2 \).
04

Compute Each Term

Evaluate each term separately: \( a^2 = x^2 \) \( 2ab = 2(x)(8) = 16x \) \(b^2 = 8^2 = 64 \)
05

Write the Final Expression

Combine all the terms to get the expanded form: \( x^2 + 16x + 64 \).

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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 a method used to expand expressions raised to a power, where the expression consists of two terms. For example, expanding \( (x+8)^2 \) means finding an equivalent expression in a simplified form. Binomial expansion relies on the Binomial Theorem which provides a systematic way to expand binomials.
Algebraic Expressions
In algebra, an algebraic expression represents a mathematical phrase that can include numbers, variables, and operators (such as +, -, *, and /). For example, \( x+8 \) is an algebraic expression where \( x \) is a variable and \( 8 \) is a constant. When we square the expression \( (x+8) \), we are essentially performing operations on these terms according to specific rules.
Polynomial Operations
Polynomial operations involve performing addition, subtraction, multiplication, and sometimes division on polynomials. A polynomial is an expression with multiple terms, such as \( x^2 + 16x + 64 \). To expand a binomial like \[ (x+8)^2 \], you multiply the binomial by itself, and then simplify using polynomial operations. This simplification involves distributing each term and combining like terms.
Quadratic Expressions
Quadratic expressions are polynomials of degree 2, which means the highest power of the variable is 2. The expanded form of \( (x+8)^2 \) is a quadratic expression since it simplifies to \[ x^2 + 16x + 64 \]. You can identify a quadratic expression by its standard form \[ ax^2 + bx + c \], where \[ a, b, \] and \[ c \] are constants. Understanding quadratic expressions is pivotal for solving various algebraic problems, including factoring and solving quadratic equations.

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Most popular questions from this chapter

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

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ can be used to perform some multiplication problems. Here are two examples. $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1 \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 30 \frac{1}{3} \times 29 \frac{2}{3} $$

A computer can perform \(466,000,000\) calculations per second. How many calculations can it perform per minute? Per hour?

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 y^{-1} z^{2}\right)^{2}\left(3 y^{-2} z^{-3}\right)^{3}}{\left(y^{3} z^{2}\right)^{-1}} $$

Match each number written in scientific notation I with the correct choice from Column II. Not all choices in Column II will be used. (I) (a) \(1 \times 10^{9}\) (b) \(1 \times 10^{6}\) (c) \(1 \times 10^{8}\) (d) \(1 \times 10^{10}\) (II) A. 1 billion B. 100 million C. 1 million D. 10 billion E. 100 billion
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