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Chapter 9: Problem 51

Expand and multiply. $$(2 x-4)^{2}$$

Short Answer

Expert verified
The expanded form is \(4x^2 - 16x + 16\).

Step by step solution

01

Recognize the Formula

To expand the expression \((2x - 4)^2\), recognize it as a perfect square. Recall the formula for a square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 2x\) and \(b = 4\).
02

Apply the Formula

Substitute \(a = 2x\) and \(b = 4\) into the formula: \((2x - 4)^2 = (2x)^2 - 2(2x)(4) + (4)^2\).
03

Simplify Each Term

Calculate each term separately:- First term: \((2x)^2 = 4x^2\).- Second term: \(-2(2x)(4) = -16x\).- Third term: \((4)^2 = 16\).
04

Write the Expanded Expression

Combine the simplified terms from Step 3 to form the expanded expression: \(4x^2 - 16x + 16\).

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Key Concepts

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

Perfect Squares
A perfect square is an expression obtained by multiplying a binomial by itself. This concept is central in algebra because it allows us to simplify expressions and solve equations easily. Consider the binomial
  • When you square it, you apply the identity
    • For any two terms, say \(a\) and \(b\), the formula for a perfect square is
      • Applying this to the original exercise, the binomial \((2x - 4)\) is squared to become \((2x - 4)^2\).
      • This means we can expand the expression using the formula:
        • where \(a = 2x\) and \(b = 4\).
        Through understanding perfect squares, you can recognize patterns and expand expressions with ease, knowing these will always result in three specific terms: the square of the first term, twice the product of both terms, and the square of the second term.
Polynomials
Polynomials are expressions composed of variables, coefficients, and the arithmetical operations of addition, subtraction, multiplication, and non-negative integer exponents. They are fundamental in algebra because they represent general forms that can be manipulated, combined, or factored in various ways.
  • In our exercise, the expanded form \(4x^2 - 16x + 16\) is a polynomial of degree two, also known as a quadratic polynomial.
  • Each term of the polynomial has specific components:
    • The first term, \(4x^2\), is the leading term, giving the polynomial its degree.
    • The middle term, \(-16x\), involves a reduction of the expression, often involving linear terms that can influence the slope or shifts in equations.
    • The last term, \(16\), is a constant term that typically affects the vertical position on a graph.
    Knowing how to expand binomials into polynomials is crucial in understanding more complex algebraic problems. It allows for simplification and reinterpretation that are essential in solving equations or inequalities.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and arithmetic operations, forming the basis of algebra. They can vary in complexity from simple monomials to complex polynomials.
  • Understanding and manipulating them, like in the expression \((2x - 4)^2\), helps develop skills in simplification and equation solving.
  • Key operations include:
    • Recognizing patterns such as perfect squares.
    • Using known formulas and identities for expansion.
    • Applying arithmetic to simplify each step.
    These operations enhance your ability to transform expressions into a more workable or simpler form.
  • Mastery of algebraic expressions lays the groundwork for higher algebraic functions and calculus. It is essential in various mathematical contexts, including analytical geometry and mathematical modeling.
By understanding how to expand and recognize forms in algebraic expressions, students can build a solid foundation for their progression in mathematics.

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