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Precalculus 9-12 / Week 02 / Monday
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Week 02 · Every Polynomial Factors

Monday

The square that would not factor
// Planks in Pascal's rows
⏱ about 20 min

Monday: The Square That Would Not Factor

The hull lies upside down on two sawhorses, its planks stacked in triangular rows while the glue sets.

"While we wait," Wren says, "the chalkboard still has last year's list. x² - 4 factors. x² + 4 does not."

"Says who?" Comet asks. "What can we make of x² + 4?"

"A difference of squares needs a minus," Wren says. "x² - 4 is (x + 2)(x - 2). There is no minus in x² + 4."

Nova projects last week's grid. "Would you like a hint? What is (2i)²?"

"i² is -1, so (2i)² is -4," Comet says. "Then x² + 4 is x² minus (2i)². A difference of squares after all!"

"(x + 2i)(x - 2i)," Wren writes, and checks it. "The i terms cancel and the end is -(2i)², which is 4."

An upturned hull with planks in triangular stacked rows; Comet fits a plank while Wren counts the rows.

An old identity, a bigger number system

In Algebra 2 the difference of squares a² - b² = (a + b)(a - b) worked for real numbers. It still works when a or b is complex.

Since (2i)² = -4, the sum x² + 4 is really x² - (2i)². So x² + 4 = (x + 2i)(x - 2i).

Check by multiplying: x² - 2ix + 2ix - (2i)² = x² - (-4) = x² + 4. The middle terms cancel.

Every polynomial identity you know extends this way. The complex numbers follow the same rules of arithmetic as the reals.

A solved problem to study

  1. Factor x² + 9 over the complex numbers.
  2. Find a number whose square is -9: (3i)² = -9.
  3. Rewrite the sum as a difference: x² + 9 = x² - (3i)².
  4. Apply a² - b² = (a + b)(a - b) with a = x and b = 3i. The factors are (x + 3i)(x - 3i).
  5. Check: the middle terms cancel and -(3i)² = -(-9) = 9. True.

Notice the pattern. Every x² + c with c positive is x² minus the square of √c times i. So every one of them factors.

Powers of i, reviewed

PowerValueWhy
i¹ithe start
i²-1the definition of i
i³-ii² × i
i⁴1i² × i²
(2i)²-44 × i²
(3i)²-99 × i²
FACTOR OVER THE COMPLEX NUMBERS
  • Read the question.
  • Tap your answer.
Factor x² + 4 over the complex numbers.
The chalkboard's next line is x² + 9. Factor it over the complex numbers.
Factor x² + 1 over the complex numbers.
Factor x² + 25 over the complex numbers.
POWERS AND IDENTITIES
  • Read the question.
  • Tap your answer.
What is i²?
What is i³?
The real-number version first: which expression equals x² - 9?
x² + 16 factors as (x + 4i)(x - ?i). What number goes in the blank? Type the number.
WHY THIS EXERCISEx² + c always splits into two conjugate factors, one with +i and one with -i.
StatementTrue or false?
x² + 4 = (x + 2i)(x - 2i).?
x² + 4 = (x + 2)(x - 2).?
(2i)² = -4.?
The identity a² - b² = (a + b)(a - b) stops working when b is imaginary.?
4 × 4 = 16?
WHY THIS EXERCISESeeing x² + 4 as a difference of squares is the first step toward factoring every polynomial.
Try it
On an index card, write x² + 1, x² + 4, x² + 9 and x² + 16 down one side.
Write each factored form on the other side. Multiply one back out to check that the middle terms cancel.
Draw the two zeros of x² + 4, which are 2i and -2i, on a complex plane. Mark their reflection in the real axis.

Strong start. Tomorrow the big theorem: every polynomial has exactly as many zeros as its degree.