# Lesson 6

Different Forms

- Let’s use the different forms of polynomials to learn about them.

### 6.1: Which One Doesn’t Belong: Small Differences

Which one doesn’t belong?

A: \(y=(x+4)(x-6)\)

B: \(y=2x^2-8x-24\)

C: \(y=x^2+5x-25\)

D: \(y=x^3+3x^2-10x-24\)

### 6.2: The Return of the Box

Earlier, we learned we can make a box from a piece of paper by cutting squares of side length \(x\) from each corner and then folding up the sides. Let’s say we now have a piece of paper that is 8.5 inches by 14 inches. The volume \(V\), in cubic inches, of the box is a function of the side length \(x\) where \(V(x)=(14-2x)(8.5-2x)(x)\).

- Identify the degree and leading term of the polynomial. Explain or show your reasoning.
- Without graphing, what can you say about the horizontal and vertical intercepts of the graph of \(V\)? Do these points make sense in this situation?

### 6.3: Using Diagrams to Multiply

- Use the distributive property to show that each pair of expressions is equivalent.
- \((x+2)(x+4)\) and \(x^2 + 6x + 8\)
- \((x+6)(x+\text-5)\) and \(x^2 + x -30\)
- \((x^2+10x+7)(2x-1)\) and \(2x^3+19x^2+4x-7\)
- \((4x^3-8)(x^2+3)\) and \(4x^5+12x^3-8x^2-24\)

- Write a pair of expressions that each have 2 or 3 terms, and trade them with your partner. Multiply the expressions they gave you.

### 6.4: Spot the Differences

Let \(f(x)=(x-2)(x+3)(x-7)\) and \(g(x)=\frac12 (x-2)(x+3)(x-7)\).

- Use the applet to explore both functions in the same window of \(\text-10 \leq x \leq 10\) and \(\text-100 \leq y \leq 100\). Describe how the two graphs are the same and how they are different.
- What degree do these polynomials have? Rewrite each expression in standard form to check.
- Let \(h(x)= (3x-6)(x+3)(x-7)\). What do you think the graph of \(y=h(x)\) will look like compared to \(y=f(x)\)? Use the applet to check your prediction.

Here are the graphs of two polynomial functions, \(f\) and \(g\). We know that \(g(x)=k\boldcdot f(x)\).

- Why do the two graphs have different vertical intercepts but the same horizontal intercepts?
- What is the value of \(k\)?

### Summary

We can express polynomials in different, equivalent, algebraic forms. These forms can give us different information about features of the polynomial and its graph. Earlier, we learned about expressing a polynomial function in factored form to identify zeros. The standard form of a polynomial, that is, the expanded version of factored form, makes it easier to identify different information about a polynomial.

For example, here are the expressions for a polynomial \(P(x)\) written in factored form and standard form:

\(\displaystyle P(x) = 0.25(x-1)^2(x+2)(x-3)(x+3)\)

\(\displaystyle P(x) = 0.25x^5 - 3x^3 + 0.5x^2 + 6.75x - 4.5\)

In standard form, two key features of the polynomial function can be identified: the constant term and the degree.

The constant term, shown as -4.5 in the example, tells us the value of the function when \(x=0\). In a graph of the function, this point is known as the vertical intercept.

The degree, shown as 5 in the example, tells us about the general shape of the graph of the polynomial, which is something we’ll learn more about in future lessons.