When given zeros to find a function, we need to remember that factors multiply to give us our polynomial. For example, if we have the factors 6 and 2, they evenly divide into 12, meaning 6 times 2 gives us 12. The same applies to our polynomial. If we’re given zeros, we need to write them as factors.

To write zeros as factors, we need to remember how we go from factors to zeros. For instance, if we have x² + 9 and x – 1 = 0, we set the zero product property and solve for zero, which gives us x = 1. If we have a square, we have x² = -9, then we square root and get x = ± √(-9). If given one complex number as a root, we need to include the opposite value because when we take the square root to find the zero, it’s always going to be plus or minus.

Now that we have our zeros, we can write them as a series of factors. For example, x – 4, x + 3i, and x – 3i. To get x – 4, we set the x-intercept equal to 4, then subtract the 4 on the other side.

To multiply our factors and find our polynomial, we can use the difference of two squares to make it quicker. For instance, if our first two terms are the same as the last two terms, we can rewrite them as x² – 3i². Since i² is -1, -3i² equals +3. Then, we can multiply the two binomials to get our polynomial, which will be x³ – 4x² + 9x – 36.

In summary, when given zeros, we need to write them as factors to find our polynomial. We can use the zero product property to solve for zero, then multiply our factors using the difference of two squares.