Question

Write the following rational expression in lowest terms.
StartFraction 1 minus 64 r cubed Over 32 r squared plus 8 r plus 2 EndFraction

Answer 1

To simplify the rational expression \(\frac{1 - 64r^3}{32r^2 + 8r + 2}\), we will start by factoring both the numerator and the denominator.

1. **Factoring the Numerator:**
\[
1 - 64r^3 = 1 - (4r)^3
\]
This expression is a difference of cubes, and it can be factored using the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\):
\[
1 - (4r)^3 = (1 - 4r)((1)^2 + (1)(4r) + (4r)^2) = (1 - 4r)(1 + 4r + 16r^2)
\]

2. **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:
\[
32r^2 + 8r + 2 = 2(4r + 1)^2
\]

3. **Putting It All Together:**
The expression can now be rewritten as:
\[
\frac{(1 - 4r)(1 + 4r + 16r^2)}{2(4r + 1)^2}
\]

4. **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:
\[
\frac{(1 - 4r)(1 + 4r + 16r^2)}{2(4r + 1)^2}
\]