1. **State the problem:** We want to find the value of $z$ that makes each polynomial a perfect trinomial square. 2. **Recall the form of a perfect square trinomial:** A perfect square trinomial looks like $a^2 + 2ab + b^2 = (a+b)^2$. 3. **Part (a):** $r^2 + zr + 100$ - Here, $a^2 = r^2$ so $a = r$. - The constant term $b^2 = 100$ so $b = 10$ (taking positive root for simplicity). - The middle term should be $2ab = 2 \times r \times 10 = 20r$. - Therefore, $z = 20$ to make it a perfect square. 4. **Part (b):** $9k^2 - 24k + z$ - Here, $a^2 = 9k^2$ so $a = 3k$. - The middle term is $-24k$, which should equal $2ab = 2 \times 3k \times b = 6kb$. - Equate $6kb = -24k$ to find $b$: $6b = -24 \Rightarrow b = -4$. - The constant term $b^2 = (-4)^2 = 16$, so $z = 16$. **Final answers:** - (a) $z = 20$ - (b) $z = 16$