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 \(V(x)=(14-2x)(8.5-2x)(x)\).

  1. Identify the degree and leading term of the polynomial. Explain or show your reasoning.
  2. 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

  1. Use the distributive property to show that each pair of expressions is equivalent.
    1. \((x+2)(x+4)\) and \(x^2 + 6x + 8\)
    2. \((x+6)(x+\text-5)\) and \(x^2 + x -30\)
    3. \((x^2+10x+7)(2x-1)\) and \(2x^3+19x^2+4x-7\)
    4. \((4x^3-8)(x^2+3)\) and \(4x^5+12x^3-8x^2-24\)
  2. 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 \(g(x)=\frac12 (x-2)(x+3)(x-7)\).

  1. 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.
  2. What degree do these polynomials have? Rewrite each expression in standard form to check.
  3. 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 \(g(x)=k\boldcdot f(x)\).

  1. Why do the two graphs have different vertical intercepts but the same horizontal intercepts?
  2. What is the value of \(k\)?
Two curves on a coordinate plane.  X-intercepts are the same for both functions. F of x has more extreme maximums and minimum than G of x. F of 0 is negative 5, G of 0 is negative 1.

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.