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Algebra 2 9-12 / Week 02 / Monday
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Week 02 · Polynomial Functions

Monday

The planter box
// A box whose volume is a cubic
⏱ about 20 min

Monday: The Planter Box

Comet kneels on the Glass House floor beside a flat sheet of cardboard. "Twenty-four by eighteen centimeters. We cut a square from each corner and fold."

"How big a square?" Wren asks, pencil ready.

"Call the side x," Comet says. "What can we make? A box that holds the most seedling soil."

Wren sketches the fold. "Then the base is 24 - 2x by 18 - 2x, and the height is x. What do you notice?"

"Volume is length times width times height," Comet says. "(24 - 2x)(18 - 2x)x."

Nova projects the expression and lets it expand. "Would you like a hint? Count the highest power."

"x times x times x," Wren says. "A cubic. Our first polynomial of degree three."

He draws a wavy curve on graph paper. "Somewhere on this curve is the best cut."

Comet measures a half-built cardboard planter box while Wren draws a wavy curve on graph paper and Nova hovers.

A polynomial function

A polynomial function adds terms, each a number times a whole-number power of x. The largest power is the degree.

The crew's box has volume V(x) = (24 - 2x)(18 - 2x)x. Expanded, V(x) = 4x³ - 84x² + 432x.

Its degree is 3, so it is a cubic. The number in front of x³, which is 4, is the leading coefficient.

Lines are degree 1 and quadratics are degree 2. You know those. This week the degree goes higher.

A solved problem to study

How much does the box hold when the cut is 3 centimeters? Read the two ways, then compare.

  1. Way one, factored form: V(3) = (24 - 6)(18 - 6)(3) = 18 × 12 × 3 = 648.
  2. Way two, expanded form: V(3) = 4(27) - 84(9) + 432(3) = 108 - 756 + 1296 = 648.
  3. Both forms are the same function, so both give the same volume, in cubic centimeters.
  4. The factored form shows the build: length, width, height. The expanded form shows the degree and the leading term.

Both forms are useful, and this week you move between them. The factored form is quickest for zeros.

The expanded form is quickest for end behavior. Keep both in the log.

The crew's cut table

Every value is the crew's own, worked from their sheet. Volumes are in cubic centimeters.

Cut xBaseHeightVolume V(x)
122 by 161352
220 by 142560
318 by 123648
416 by 104640
514 by 85560
612 by 66432
710 by 47280
88 by 28128

The volume rises, peaks, then falls. In the table the largest whole-number cut is x = 3, with V(3) = 648.

EVALUATE THE BOX FUNCTION
  • Read the question.
  • Tap your answer.
If V(x) = 4x³ - 84x² + 432x, what is V(2)?
If V(x) = 4x³ - 84x² + 432x, what is V(3)?
If V(x) = 4x³ - 84x² + 432x, what is V(4)?
What is the degree of V(x) = 4x³ - 84x² + 432x?
In the crew's table, which whole-number cut gives the largest volume? Type the number.
WHY THIS EXERCISEA polynomial can rise and then fall. Finding where it peaks is the point of the build.
StatementTrue or false?
(24 - 2 × 3) × (18 - 2 × 3) × 3 = 648?
(24 - 2 × 2) × (18 - 2 × 2) × 2 = 560?
V(x) = 4x³ - 84x² + 432x has degree 4.?
The factored and expanded forms of V(x) give the same value for every x.?
The leading coefficient of V(x) is 432.?
WHY THIS EXERCISEReading degree and leading coefficient correctly is the first step in sketching any polynomial.
Try it
Cut four 1-centimeter squares from the corners of an index card with scissors. Fold up the sides and tape them.
Measure the base and height. Multiply to find the volume, then check it with your card's own V(x).
Draw the flat sheet with its four corner squares marked x, then the folded box with its base and height labeled.

A solid start. Tomorrow you sketch polynomials from their zeros and end behavior, two features at a time.