To find the Greatest Common Factor (GCF) of the polynomial \(4x^4 + 12x^3 - 36x^2 + 24x\), we will first factor out any common factors from each term.
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Break down each term:
- \(4x^4\): has a factor of \(4\) and \(x^4\)
- \(12x^3\): has a factor of \(12\) and \(x^3\)
- \(-36x^2\): has a factor of \(-36\) and \(x^2\)
- \(24x\): has a factor of \(24\) and \(x\)
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Identify the GCF of the coefficients:
- The coefficients are \(4\), \(12\), \(-36\), and \(24\).
- The GCF of \(4\), \(12\), \(36\), and \(24\) is \(12\).
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Identify the GCF of the variable parts:
- The variable parts are \(x^4\), \(x^3\), \(x^2\), and \(x\).
- The GCF of \(x^4\), \(x^3\), \(x^2\), and \(x\) is \(x\).
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Now, combine the GCF of the coefficients and the variables, which gives: \[ GCF = 4x \]
Thus, the GCF of the polynomial \(4x^4 + 12x^3 - 36x^2 + 24x\) is \(4x\).