1. **Stating the problem:** We have a polynomial function $f(x) = 5x^5 + bx^4 + cx^3 + dx^2 - 5$ and we know that $x=\frac{1}{2}$ is a root of this polynomial. We want to find the value of the expression $16b + 8c + 4d$. 2. **Using the root condition:** Since $x=\frac{1}{2}$ is a root, substituting $x=\frac{1}{2}$ into $f(x)$ must give zero: $$ 5\left(\frac{1}{2}\right)^5 + b\left(\frac{1}{2}\right)^4 + c\left(\frac{1}{2}\right)^3 + d\left(\frac{1}{2}\right)^2 - 5 = 0 $$ 3. **Calculate powers of $\frac{1}{2}$:** $$ \left(\frac{1}{2}\right)^5 = \frac{1}{32}, \quad \left(\frac{1}{2}\right)^4 = \frac{1}{16}, \quad \left(\frac{1}{2}\right)^3 = \frac{1}{8}, \quad \left(\frac{1}{2}\right)^2 = \frac{1}{4} $$ 4. **Substitute these values:** $$ 5 \times \frac{1}{32} + b \times \frac{1}{16} + c \times \frac{1}{8} + d \times \frac{1}{4} - 5 = 0 $$ 5. **Simplify the constants:** $$ \frac{5}{32} + \frac{b}{16} + \frac{c}{8} + \frac{d}{4} - 5 = 0 $$ 6. **Isolate terms with $b,c,d$:** $$ \frac{b}{16} + \frac{c}{8} + \frac{d}{4} = 5 - \frac{5}{32} = \frac{160}{32} - \frac{5}{32} = \frac{155}{32} $$ 7. **Multiply both sides by 16 to clear denominators:** $$ 16 \times \left(\frac{b}{16} + \frac{c}{8} + \frac{d}{4}\right) = 16 \times \frac{155}{32} $$ $$ b + 2c + 4d = \frac{2480}{32} = 77.5 $$ 8. **Express $16b + 8c + 4d$ in terms of $b,c,d$:** Note that $$ 16b + 8c + 4d = 16b + 8c + 4d $$ 9. **Relate $16b + 8c + 4d$ to $b + 2c + 4d$:** Multiply the equation $b + 2c + 4d = 77.5$ by 16: $$ 16b + 32c + 64d = 16 \times 77.5 = 1240 $$ 10. **We want $16b + 8c + 4d$, so subtract $24c + 60d$ from both sides:** $$ 16b + 8c + 4d = 1240 - 24c - 60d $$ Without additional information about $c$ and $d$, we cannot simplify further. However, the problem likely meant $16b + 8c + 4d$ equals the value found by substituting $x=\frac{1}{2}$, so we can rewrite the original substitution step: From step 6: $$ \frac{b}{16} + \frac{c}{8} + \frac{d}{4} = \frac{155}{32} $$ Multiply both sides by 16: $$ b + 2c + 4d = \frac{155}{32} \times 16 = \frac{155}{2} = 77.5 $$ Multiply this entire equation by 16: $$ 16b + 32c + 64d = 16 \times 77.5 = 1240 $$ We want $16b + 8c + 4d$, which is less than $16b + 32c + 64d$. Without more info, the problem as stated is ambiguous. Assuming a typo and the problem meant $16b + 8c + 4d$ equals the value from the substitution, then: Multiply the original substitution equation by 16: $$ 16 \times \left(\frac{b}{16} + \frac{c}{8} + \frac{d}{4}\right) = 16 \times \frac{155}{32} $$ $$ b + 2c + 4d = 77.5 $$ Multiply both sides by 16: $$ 16b + 32c + 64d = 1240 $$ Since the problem asks for $16b + 8c + 4d$, and we have $16b + 32c + 64d$, we cannot find a unique value without more info. **Final answer:** The value of $16b + 8c + 4d$ cannot be determined uniquely from the given information.