To simplify the polynomial expression \( 6x^2 - 4 \cdot x^2 - 164x + 16 \), let's first combine like terms.
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Combine \( 6x^2 \) and \( -4x^2 \): \[ 6x^2 - 4x^2 = 2x^2 \] So now we have: \[ 2x^2 - 164x + 16 \]
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Next, let's factor out the greatest common factor (GCF) from the polynomial \( 2x^2 - 164x + 16 \): The GCF of the coefficients \( 2, -164, 16 \) is \( 2 \). Factoring out \( 2 \) gives: \[ 2(x^2 - 82x + 8) \]
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Now, we need to factor the quadratic \( x^2 - 82x + 8 \). To factor this, we can look for two numbers that multiply to \( 8 \) and add to \( -82 \). However, since this won’t yield a simple factorization, we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1, b = -82, c = 8 \): \[ x = \frac{82 \pm \sqrt{(-82)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} = \frac{82 \pm \sqrt{6724 - 32}}{2} = \frac{82 \pm \sqrt{6692}}{2} \]
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Since \( 6692 \) cannot be simplified easily and doesn't yield simple integer factors, we will leave the expression at our GCF factoring stage.
Thus, the simplest polynomial form is: \[ \frac{2(x^2 - 82x + 8)}{1} \]
The closest response from your list that matches is:
- \( 3x(x+4)(x-4) \) This is incorrect as it does not match the output of our polynomial.
None of the options you provided perfectly simplify or accurately capture the correct output. Thus, based on the calculations done, the option for \( 2(x^2 - 82x + 8) \) can be taken as the polynomial’s simplest form, if expressed correctly or through the GCF method.
If you are looking for an exact among the provided options, please ensure that any context in assessment allows re-evaluation of polynomial expressions or possible errors in options provided here to interpret the proper quadratic expression.
3 answers