Find the value of C = 8x³ + 12x³ + 6x + 1 at x = 1/2.

Step-by-step solution

1. 1

   Combine the like terms first. Both leading terms carry $x^{3}$, so they add directly:

   $8x^{3}+12x^{3}=20x^{3},\qquad C=20x^{3}+6x+1.$

   Simplifying before substituting means cubing $\tfrac12$ once rather than twice.

2. 2

   Substitute $x=\tfrac12$ into the cubic term.

   $\left(\frac12\right)^{3}=\frac18,\qquad 20\cdot\frac18=\frac{20}{8}=\frac52.$

3. 3

   Substitute into the linear term.

   $6\cdot\frac12=3.$

4. 4

   Add the three contributions.

   $C=\frac52+3+1=\frac52+4=\frac52+\frac82=\frac{13}{2}=6.5.$

5. 5

   Watch the near-miss. If the middle term had been $12x^{2}$ rather than $12x^{3}$, the expression would be

   $8x^{3}+12x^{2}+6x+1=(2x+1)^{3},$

   a perfect cube, and its value at $x=\tfrac12$ would be $(1+1)^{3}=8$, not $\tfrac{13}{2}$. The two differ, so the exponent on the second term must be read carefully before any factoring shortcut is applied.