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Chapter 9: Problem 128

Simplify. $$(x+4)^{3}$$

Short Answer

Expert verified
The simplified version of \((x + 4)^3\) is \(x^3 + 12x^2 + 48x + 64\).

Step by step solution

01

Recognize the pattern

We're working with the situation \((x + 4)^3\). The power is 3, meaning we're dealing with a cubic pattern. In general terms, \((a+b)^3=a^3+3a^2b+3ab^2+b^3 \). Here a is x and b is 4.
02

Apply the binomial theorem

To expand \((x + 4)^3\), we apply the formula given above, substitute x for a and 4 for b. We get \(x^3 + 3 * x^2 * 4 + 3 * x * 4^2 + 4^3\).
03

Simplify

Next, we simplify the equation, and we get \(x^3 + 12x^2 + 48x + 64\).

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Key Concepts

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

Cubic Expansion
Cubic expansion is a method used to expand expressions raised to the third power, like \((x + 4)^3\). It involves the **Binomial Theorem**, which provides a formula for expanding powers of binomials. The formula \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)is essential.
  • **First Term**: Cube the first term \(a\). So, \(a^3\).
  • **Second Term**: Multiply the square of \(a\) by \(b\) and by 3, resulting in \(3a^2b\).
  • **Third Term**: Multiply \(a\) by the square of \(b\) and by 3, leading to \(3ab^2\).
  • **Fourth Term**: Cube the second term \(b\), resulting in \(b^3\).
In this example, substituting \(a = x\) and \(b = 4\) into the formula helps you discover all parts needed for full expansion. Understanding cubic expansion is crucial for handling higher power binomials effectively.
Algebraic Expressions
Algebraic expressions are made up of numbers, variables, and operations.These expressions allow us to represent real-life situations using mathematical language.An expression like \(x^3 + 12x^2 + 48x + 64\)results from simplifying a binomial expansion. Algebraic terms are defined by:
  • **Coefficients**: Numbers in front of variables, like 12 in \(12x^2\).
  • **Variables**: Letters representing unknowns, such as \(x\).
  • **Constants**: Regular numbers like 64.
Understanding how to organize and manipulate these elements is essential in algebra. They form the basis for equations,which can be solved to find unknown values. Recognizing these elements in an expression helps in breaking it downstep by step, making the problem easier to solve or simplify.
Simplifying Expressions
Simplifying expressions means making them easier to understand or work with. For the cubic expansion of \((x + 4)^3\), simplifying involves calculating and combining like terms.The process requires:
  • Calculating powers of numbers: \(4^2 = 16\) and \(4^3 = 64\).
  • Multiplying factors: Finding \(3 \times 4^2 \times x\) results in \(48x\).
  • Combining like terms: Organizing \(x^3, 12x^2, 48x,\) and \(64\) into a simplified format.
Simplifying is crucial for solving algebra problems and understanding what an expression truly represents. It allows students to focus on the structure and meaning, reducing complex problems into more manageable parts. Mastering this skill is fundamental for progressing in algebra and other math-related subjects.

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