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How to Factorize the Algebraic Expression $3x^2 - 12$
Mathematics
Grade 8 of Junior High School
Question Content
Factorize the algebraic expression: $3x^2 - 12$
Correct Answer
$3(x+2)(x-2)$
Detailed Solution Steps
1
Step 1: Identify and factor out the greatest common factor (GCF) of the two terms. The GCF of $3x^2$ and $12$ is 3, so we rewrite the expression as $3(x^2 - 4)$.
2
Step 2: Recognize that $x^2 - 4$ is a difference of perfect squares, which follows the formula $a^2 - b^2 = (a+b)(a-b)$. Here, $a=x$ and $b=2$ since $x^2 = x^2$ and $4=2^2$.
3
Step 3: Apply the difference of squares formula to factor $x^2 - 4$ into $(x+2)(x-2)$. Combine it with the factored-out GCF to get the final fully factored form: $3(x+2)(x-2)$.
Knowledge Points Involved
1
Greatest Common Factor (GCF) for Polynomials
The GCF of a polynomial is the largest expression that divides all terms of the polynomial evenly. Factoring out the GCF is the first step in most polynomial factorization problems, simplifying the expression for further factoring. For example, in $3x^2 -12$, 3 divides both $3x^2$ and 12, so it is the GCF.
2
Difference of Perfect Squares
A difference of perfect squares is a binomial of the form $a^2 - b^2$, where $a$ and $b$ are real numbers or algebraic expressions. It can be factored into the product of two binomials: $(a+b)(a-b)$. This formula only works for subtraction of two perfect squares, not addition.
3
Polynomial Factorization
Factorization is the process of breaking down a polynomial into a product of simpler polynomials (factors) that multiply together to give the original polynomial. The goal is to write the polynomial in its fully factored form, where no further factoring can be done.
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