Comet holds up two cards for the arch. "-x² + 6x - 5. And -(x - 5)(x - 1). Same rule, two outfits."
"Which one tells you where the feet are?" Wren asks. "The factored one. Zeros at 1 and 5, no work."
"And which tells you the height at x = 0?" Comet reads the standard form. "The last term, -5. Below the table."
Nova projects a square frame with side x + 3 and a border cut away. "Would you like a hint? Which form fits each question?"
"Area of the whole frame: (x + 3) squared. Expand it when I need the pieces," Comet says.
"Factor when you need zeros, expand when you need a value or a shape," Wren says. "Choose the form for the question."
"Our engineer," Comet says, "sort today's questions by form, and then let us review the week."
Standard form, -x² + 6x - 5, shows the value at x = 0 at a glance: the last term, -5. It also shows the direction, down, from the negative x² term.
Factored form, -(x - 5)(x - 1), shows the zeros at a glance: 1 and 5.
Both forms are the same polynomial. You choose the form that answers your question, then rewrite to get there.
| Question | Best form | Why |
|---|---|---|
| Where does the arch meet the table? | factored | the zeros are the numbers in the factors |
| What is the height at x = 0? | standard | the last term is the value at zero |
| Does it open up or down? | standard | the sign of the x² term says |
| Where is the top? | factored | halfway between the zeros |
| Statement | True or false? |
|---|---|
| -x² + 6x - 5 = -(x - 1)(x - 5) | ? |
| x² - 16 = (x - 4)(x + 4) | ? |
| x² + 9x + 20 = (x + 4)(x + 5) | ? |
| x² - 2x - 15 = (x - 3)(x + 5) | ? |
| The factored form and the standard form of a polynomial are equal for every x. | ? |
Terrific week. You can factor four ways, read zeros from factors and pick the form that answers the question. Tomorrow is Loft Day.