Math Problem Statement

次の式を展開せよ。 (x - 4)^3 = x^3 - アイ x^2 + ウエ x - オカ

Solution

The problem asks to expand (x4)3(x - 4)^3 and identify the coefficients of the terms in the expanded form.

We can use the binomial expansion formula to expand (x4)3(x - 4)^3:

(ab)3=a33a2b+3ab2b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

In this case, a=xa = x and b=4b = 4, so:

(x4)3=x33x2(4)+3x(42)43(x - 4)^3 = x^3 - 3x^2(4) + 3x(4^2) - 4^3

Now, calculate each term:

  1. x3x^3 remains as x3x^3.
  2. 3x2(4)=12x2-3x^2(4) = -12x^2.
  3. 3x(42)=3x(16)=48x3x(4^2) = 3x(16) = 48x.
  4. 43=64-4^3 = -64.

Thus, the expanded form is:

x312x2+48x64x^3 - 12x^2 + 48x - 64

From the expression in the image, the boxed values are:

  • アイ = 1212
  • ウエ = 4848
  • オカ = 6464

Let me know if you'd like further clarification on this, or if you have other questions!


Here are some related questions to consider:

  1. How does the binomial expansion formula work for larger powers?
  2. What is the general form for (xa)n(x - a)^n?
  3. How do we expand expressions of the form (ax+b)n(ax + b)^n?
  4. What other methods can we use to expand (x4)3(x - 4)^3?
  5. How would the coefficients change if we expanded (x+4)3(x + 4)^3?

Tip: Always check for patterns in binomial expansions—they can simplify complex terms into recognizable sequences!

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Math Problem Analysis

Mathematical Concepts

Binomial Expansion
Cubic Equations
Polynomial Expansion

Formulas

(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

Theorems

Binomial Theorem

Suitable Grade Level

Grades 8-10