To find the greatest common factor (GCF) of the polynomial \( 6x^3 - 12x^2 + 18x \), we will first identify the coefficients and the variables in each term:
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The terms are:
- \( 6x^3 \)
- \( -12x^2 \)
- \( 18x \)
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Now, let's break down the coefficients:
- The coefficients are \( 6 \), \( -12 \), and \( 18 \).
- The GCF of \( 6, -12, \) and \( 18 \) can be found by looking at the numerical factors:
- The prime factorization of \( 6 \) is \( 2 \cdot 3 \).
- The prime factorization of \( -12 \) (ignoring the sign for factorization) is \( 2^2 \cdot 3 \).
- The prime factorization of \( 18 \) is \( 2 \cdot 3^2 \).
- The GCF of the coefficients is \( 6 \) (the largest factor common to all three coefficients).
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Next, we consider the variable part. The variables in each term are:
- \( x^3 \)
- \( x^2 \)
- \( x^1 \)
The GCF of the variable parts is \( x^1 \) (or simply \( x \)) since the lowest power of \( x \) present in all terms is \( x \).
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Now we combine the GCF of the coefficients and the variables:
- The GCF of the polynomial is \( 6x \).
Therefore, the GCF of the polynomial \( 6x^3 - 12x^2 + 18x \) is \( 6x \).