A polynomial has zeros at -4, 2 and 1. Which of the following could be that polyno­mial?

To determine which polynomial could have zeros at -4, 2, and 1, we first need to know the general form of a polynomial given its zeros.

Step 1: Break Down the Problem

• The zeros of the polynomial are at $x = -4$, $x = 2$, and $x = 1$.
• The general form of a polynomial based on its zeros is given by: $f(x) = k(x + 4)(x - 2)(x - 1)$ where $k$ is a constant.

Step 2: Relevant Concepts

• Each factor corresponds to a root of the polynomial; $(x + 4)$ for the root at -4, $(x - 2)$ for the root at 2, and $(x - 1)$ for the root at 1.

Step 3: Analysis and Detail

• We can expand $f(x)$:
  1. First, multiply two of the factors: $(x - 2)(x - 1) = x^2 - 3x + 2$
  2. Now multiply by the third factor: $f(x) = k(x + 4)(x^2 - 3x + 2)$
  3. Expanding further gives: $f(x) = k \left( x^3 + (4 - 3)x^2 + (8 - 12)x + 8 \right)$ $f(x) = k (x^3 + x^2 - 4x + 8)$

Step 4: Verify and Summarize

• The simplified form of the polynomial is $k (x^3 + x^2 - 4x + 8)$. By selecting a specific value for $k$, such as $k=1$, we obtain a polynomial that has the specified zeros: $f(x) = x^3 + x^2 - 4x + 8$

Final Answer

Thus, a possible polynomial with zeros at -4, 2, and 1 is: $f(x) = x^3 + x^2 - 4x + 8$ Any scalar multiple of this polynomial, formed by varying $k$, would also be a valid polynomial meeting the condition.