To factor the expression \(20r^3 - 10r^2 + 18r - 9\), we can follow these steps:
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Group the terms: The expression can be grouped into two pairs: \[ (20r^3 - 10r^2) + (18r - 9) \]
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Factor out common factors in each group: From the first group \(20r^3 - 10r^2\), we can factor out \(10r^2\): \[ 10r^2(2r - 1) \] From the second group \(18r - 9\), we can factor out \(9\): \[ 9(2r - 1) \]
Now the expression looks like this: \[ 10r^2(2r - 1) + 9(2r - 1) \]
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Factor out the common factor \((2r - 1)\): Now we see that both terms contain the common factor \((2r - 1)\): \[ (2r - 1)(10r^2 + 9) \]
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Check if \(10r^2 + 9\) can be factored further: The expression \(10r^2 + 9\) cannot be factored further using real numbers because it does not produce real roots (its discriminant \(b^2 - 4ac\) is negative).
Thus, the fully factored form of the expression \(20r^3 - 10r^2 + 18r - 9\) is: \[ \boxed{(2r - 1)(10r^2 + 9)} \]