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Answer :
To factor the expression [tex]3x^2 - 147[/tex] completely, we will follow these steps:
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]
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].
Factor the Difference of Squares:
Apply the formula to [tex]x^2 - 49[/tex]:
[tex]x^2 - 49 = (x - 7)(x + 7)[/tex]
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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