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Difficulty: EasyOperations on Polynomials

A square has a side length of x3−3x^3 - 3 inches. Which of the following expressions represents the area, in square inches, of the square?

  1. A
    x6−9x^6 - 9
  2. B
    x9−6x3+9x^9 - 6x^3 + 9
  3. x6−6x3+9x^6 - 6x^3 + 9Answer
  4. D
    x6+9x^6 + 9
  5. E
    x9−9x^9 - 9

Answer

The expression x6−6x3+9x^6 - 6x^3 + 9
To find the area of a square, square its side length: (x3−3)2(x^3 - 3)^2. Expanding this gives (x3)2−2(3)(x3)+(−3)2=x6−6x3+9(x^3)^2 - 2(3)(x^3) + (-3)^2 = x^6 - 6x^3 + 9.

Step-by-Step Solution

1
Write the formula for the area of a square.
A=s2A = s^2, where ss is the side length.
The area of a square is the side length squared.
2
Substitute the given side length s=x3−3s = x^3 - 3 into the area formula.
A=(x3−3)2A = (x^3 - 3)^2
We replace ss with the algebraic expression for the side length.
3
Expand the squared binomial (x3−3)2(x^3 - 3)^2 using the identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
(x3)2−2(x3)(3)+32=x6−6x3+9(x^3)^2 - 2(x^3)(3) + 3^2 = x^6 - 6x^3 + 9
Expanding the binomial yields the correct simplified polynomial in standard descending order.

Key Concept

Expanding a squared binomial and applying exponent rules for powers of powers.
Estimated Time:1m 0s
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Topics

Operations on Polynomials

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