Answer :

To factor the expression [tex]3x^2 - 147[/tex] completely, we will follow these steps:

  1. Identify the Greatest Common Factor (GCF):

    The GCF of the coefficients (3 and 147) is 3. Factor out the 3 from the expression:

    [tex]3(x^2 - 49)[/tex]

  2. Recognize the Difference of Squares:

    The expression inside the parentheses, [tex]x^2 - 49[/tex], is a difference of squares. A difference of squares can be factored using the formula:

    [tex]a^2 - b^2 = (a - b)(a + b)[/tex]

    In this case, [tex]a = x[/tex] and [tex]b = 7[/tex], because [tex]49 = 7^2[/tex].

  3. Factor the Difference of Squares:

    Apply the formula to [tex]x^2 - 49[/tex]:

    [tex]x^2 - 49 = (x - 7)(x + 7)[/tex]

  4. Write the Fully Factored Expression:

    Substitute back into the expression with the GCF factored out:

    [tex]3(x - 7)(x + 7)[/tex]

So, the completely factored form of the expression [tex]3x^2 - 147[/tex] is [tex]3(x - 7)(x + 7)[/tex]. This shows that by removing the GCF and using the difference of squares formula, we can completely factor the expression.

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