To simplify the rational expression \(\frac{1 - 64r^3}{32r^2 + 8r + 2}\), we will start by factoring both the numerator and the denominator.
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Factoring the Numerator: \[ 1 - 64r^3 = 1 - (4r)^3 \] This expression is a difference of cubes, and it can be factored using the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\): \[ 1 - (4r)^3 = (1 - 4r)((1)^2 + (1)(4r) + (4r)^2) = (1 - 4r)(1 + 4r + 16r^2) \]
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Factoring the Denominator: Now we will factor the denominator \(32r^2 + 8r + 2\). We notice that all coefficients can be divided by 2: \[ 32r^2 + 8r + 2 = 2(16r^2 + 4r + 1) \] Next, we factor \(16r^2 + 4r + 1\). This is a perfect square: \[ 16r^2 + 4r + 1 = (4r + 1)^2 \] Therefore, we have: \[ 32r^2 + 8r + 2 = 2(4r + 1)^2 \]
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Putting It All Together: The expression can now be rewritten as: \[ \frac{(1 - 4r)(1 + 4r + 16r^2)}{2(4r + 1)^2} \]
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Cancelling Common Factors: At this stage, we check if there are any common factors that can be canceled. The factor \((1 - 4r)\) and \((4r + 1)\) are not equivalent, and neither term from the numerator can cancel with the denominator.
Therefore, the simplified expression in lowest terms is: \[ \frac{1 - 4r}{2(4r + 1)^2}(1 + 4r + 16r^2) \]
This can be rewritten as: \[ \frac{(1 - 4r)(1 + 4r + 16r^2)}{2(4r + 1)^2} \]
Thus, the lowest terms of the rational expression is: \[ \frac{(1 - 4r)(1 + 4r + 16r^2)}{2(4r + 1)^2} \]