#### Given the expression: 3*x*^{10}−48*x*^{2}

**Final Answer**

The expression 3*x*^{10}−48*x*^{2} can be factored by first taking out the greatest common factor, which is 3*x*^{2}, giving us 3*x*^{2}(*x*^{8}−16). This is further factored into 3*x*^{2}(*x*^{2}−2)(*x*^{2}+2)(*x*^{4}+4) using the difference of squares method. The complete factorization is 3*x*^{2}(*x*^{2}−2)(*x*^{2}+2)(*x*^{4}+4).

**Explanation**

To factor the expression 3*x*^{10}−48*x*^{2}, we will follow these steps:

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

- Identify the coefficients of the terms: 3 and -48.

The greatest common factor (GCF) of 3 and 48 is 3. - Identify the variable terms:
*x*^{10}and*x*^{2}.

The GCF of these variable terms is*x*^{2}. - Now combine the GCF of the coefficients and the GCF of the variable terms.

The overall GCF is 3*x*^{2}. - Factor the GCF out of the expression:

3*x*^{10}−48*x*^{2}=3*x*^{2}(*x*^{8}−16)

**Part B: Factor the Entire Expression Completely**

- Next, we focus on factoring the expression in the parentheses:
*x*^{8}−16. - Recognize that 16is a perfect square: 16=4
^{2}, and we can express*x*^{8}as (*x*^{4})2. - Since this is a difference of squares, we can apply the difference of squares formula:

*a*2−*b*2=(*a*−*b*)(*a*+*b*)where*a*=*x*^{4}and*b*=4. - This gives us:

*x*^{8}−16=(*x*^{4}−4)(*x*^{4}+4) - To complete the factorization, notice that
*x*^{4}−4is again a difference of squares, so we can factor it further:

*x*^{4}−4=(*x*^{2}−2)(*x*^{2}+2) - Putting everything together, the completely factored form of the original expression is:

3*x*^{2}(*x*^{4}−16)=3*x*^{2}(*x*^{2}−2)(*x*^{2}+2)(*x*^{4}+4)

Thus, the complete factorization is 3*x*^{2}(*x*^{2}−2)(*x*^{2}+2)(*x*^{4}+4).

**Examples & Evidence**

An example to consider is the expression *x*^{2}−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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