Given the expression: 3×10−48×2 

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Given the expression: 3x10−48x2 

Final Answer

The expression 3x10−48x2 can be factored by first taking out the greatest common factor, which is 3x2, giving us 3x2(x8−16). This is further factored into 3x2(x2−2)(x2+2)(x4+4) using the difference of squares method. The complete factorization is 3x2(x2−2)(x2+2)(x4+4).

Explanation

To factor the expression 3x10−48x2, we will follow these steps:

Part A: Rewrite the Expression by Factoring Out the Greatest Common Factor

  1. Identify the coefficients of the terms: 3 and -48.
    The greatest common factor (GCF) of 3 and 48 is 3.
  2. Identify the variable terms: x10and x2.
    The GCF of these variable terms is x2.
  3. Now combine the GCF of the coefficients and the GCF of the variable terms.
    The overall GCF is 3x2.
  4. Factor the GCF out of the expression:
    3x10−48x2=3x2(x8−16)

Part B: Factor the Entire Expression Completely

  1. Next, we focus on factoring the expression in the parentheses: x8−16.
  2. Recognize that 16is a perfect square: 16=42, and we can express x8 as (x4)2.
  3. Since this is a difference of squares, we can apply the difference of squares formula:
    a2−b2=(a−b)(a+b)where a=x4 and b=4.
  4. This gives us:
    x8−16=(x4−4)(x4+4)
  5. To complete the factorization, notice that x4−4is again a difference of squares, so we can factor it further:
    x4−4=(x2−2)(x2+2)
  6. Putting everything together, the completely factored form of the original expression is:
    3x2(x4−16)=3x2(x2−2)(x2+2)(x4+4)

Thus, the complete factorization is 3x2(x2−2)(x2+2)(x4+4).

Examples & Evidence

An example to consider is the expression x2−9 which is factored into (x−3)(x+3), demonstrating how difference of squares work similarly to the factorization seen here.

This information is accurate as it follows the established mathematical principles for factoring polynomials, including identifying greatest common factors and applying the difference of squares formula.

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