# Write a polynomial function in factored form

Find the size of squares that should be cut out to maximize the volume enclosed by the box. In these cases, we say that the turning point is a global maximum or a global minimum. Authored by: Abramson, Jay et al. Each turning point represents a local minimum or maximum. Keep in mind that some values make graphing difficult by hand. Analysis of the Solution We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown in Figure 5.

If the polynomial function is not given in factored form: Factor out any common monomial factors. From this zoomed-in view, we can refine our estimate for the maximum volume to about cubic cm which occurs when the squares measure approximately 2.

## Factoring calculator

We will use the y-intercept 0, —2 , to solve for a. If the polynomial function is not given in factored form: Factor out any common monomial factors. Analysis of the Solution We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown in Figure 5. From this zoomed-in view, we can refine our estimate for the maximum volume to about cubic cm which occurs when the squares measure approximately 2. How To: Given a graph of a polynomial function, write a formula for the function Identify the x-intercepts of the graph to find the factors of the polynomial. In these cases, we say that the turning point is a global maximum or a global minimum. Keep in mind that some values make graphing difficult by hand. Show Solution We will start this problem by drawing a picture like the one below, labeling the width of the cut-out squares with a variable, w. Sometimes, a turning point is the highest or lowest point on the entire graph. Because a polynomial function written in factored form will have an x-intercept where each factor is equal to zero, we can form a function that will pass through a set of x-intercepts by introducing a corresponding set of factors. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus.

Solution This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed.

Technology is used to determine the intercepts. Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs.

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