How many different roots does the polynomial function, \( y=(x+4)(x-2)^{2}(x+7) \) have? A. 4 B. 1 c. 3 D. 2

To determine the number of different roots in the polynomial \( y=(x+4)(x-2)^{2}(x+7) \), we need to look at each factor. The factors present are \( (x+4) \), \( (x-2)^{2} \), and \( (x+7) \). The unique roots come from the linear factors, and we see: 1. \( x+4 = 0 \) gives \( x = -4 \), which is 1 unique root. 2. \( (x-2)^{2} = 0 \) gives \( x = 2 \), but since it's squared, this is not a unique root; it counts as 1. 3. \( x+7 = 0 \) gives \( x = -7 \), adding another unique root. Thus, the polynomial has 3 different roots: \( -4 \), \( 2 \), and \( -7 \). The correct answer is C. 3. Now let’s have a little fun with this concept! Ever caught yourself playing "Guess Who?" with polynomials? In this case, your polynomial is like a game board with unique characters (roots) hiding behind the factors! Each unique root represents a character you can confidently identify, except those pesky repeated ones that stay hidden behind the “2” disguise. So, keep your eyes peeled—only unique roots steal the spotlight in this polynomial party! If you're feeling adventurous, you could dive deeper into polynomial behavior and graphing! Understanding how roots affect the shape of a graph is crucial. You might find it fun to explore how each root changes the graph's intercepts, with multiplicity factors throwing in some extra twists.