To factor out the greatest common factor (GCF) from the polynomial \(33d^5f^9 - 19d^3f^5 + 40\), we first need to identify the GCF of the coefficients and the variables in each term.
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Look at the coefficients: \(33\), \(-19\), and \(40\).
- The GCF of \(33\), \(-19\), and \(40\) is \(1\) (since \(19\) is a prime number and does not divide \(33\) or \(40\)).
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Now look at the variable parts:
- The term \(33d^5f^9\) has \(d^5\) and \(f^9\).
- The term \(-19d^3f^5\) has \(d^3\) and \(f^5\).
- The term \(40\) has no variables.
The GCF for the variable parts, considering each variable’s exponent:
- For \(d\), the minimum exponent is \(d^0\) (since \(40\) has no \(d\)), so we factor out \(d^0 = 1\).
- For \(f\), the minimum exponent is \(f^0\) (since \(40\) has no \(f\)), so we factor out \(f^0 = 1\).
Combining these findings, the greatest common factor of the entire polynomial is \(1\).
Since the GCF is \(1\), the factored form is simply the original polynomial:
\[ 33d^5f^9 - 19d^3f^5 + 40 \]
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