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Algebra 2 9-12 / Week 04 / Tuesday
2/6
Week 04 · Rational Expressions and Equations

Tuesday

Simplify, multiply, add
// Plant food in the tank
⏱ about 20 min

Tuesday: Simplify, Multiply, Add

Rain taps the glass roof while Wren fills the chalkboard wall with fractions holding x.

"Yesterday we wrote one rational expression," he says. "Today we treat them like any fractions. What do you notice?"

Comet reads (x² - 9)/(x - 3). "The top factors. (x + 3)(x - 3) over (x - 3). Cancel, and it is x + 3."

"Like 6/3 becoming 2," Wren says. "Multiplying works the same. Factor first, then cancel across."

"And adding?" Comet asks. "We need a common denominator. Like 1/2 + 1/3 needs sixths."

Nova dims her light to a hint. "Would you like a hint? Multiply the two denominators to make one both can share."

"So 1/x + 1/(x + 1) is over x(x + 1)," Comet says. "Tops become x + 1 and x. Add: 2x + 1."

Way one: simplify by factoring

Factor the top and the bottom fully. Cancel any factor that appears in both, exactly as 6/8 becomes 3/4.

For example, (x² - 9)/(x - 3) = (x + 3)(x - 3)/(x - 3) = x + 3.

And (x² + 5x + 6)/(x² - 4) = (x + 3)/(x - 2), after canceling the shared factor x + 2.

Only whole factors cancel. Never cross out a single term from a sum, like the x in (x + 3)/x.

Way two: multiply, then simplify

To multiply, factor everything first, cancel any factor that appears on a top and a bottom, then multiply what is left.

For example, (x + 2)/(x - 1) × (x - 1)/x = (x + 2)/(x). The x - 1 cancels across.

And (x² - 4)/x × 3x/(x + 2) = 3x - 6, because x and x + 2 both cancel.

Way three: add with a common denominator

  1. Factor each denominator. The common denominator holds every factor that appears in either one.
  2. Rewrite each fraction over the common denominator by multiplying its top and bottom by the missing factor.
  3. Add or subtract the tops. Keep the common denominator.
  4. Simplify the result if the new top and bottom share a factor.
  5. So 1/x + 1/(x + 1) = (x + 1)/(x(x + 1)) + x/(x(x + 1)) = (2x + 1)/(x² + x).

Compare: multiply first, or factor first?

ProblemMultiply firstFactor firstResult
(x + 2)/(x - 1) × (x - 1)/xtop (x + 2)(x - 1), bottom x(x - 1), then cancelcancel x - 1 across, then multiply(x + 2)/(x)
(x² - 4)/x × 3x/(x + 2)a messy degree-3 top, then factor itfactor x² - 4 and cancel twice3x - 6
2/(x - 1) + 3/(x + 2)no shortcut: the bottoms share no factorcommon denominator (x - 1)(x + 2)(5x + 1)/(x² + x - 2)

Both ways give the same answer. Wren's rule: factor first whenever you can, because small factors are easier to see.

SIMPLIFY
  • Read the question.
  • Tap your answer.
Simplify (x² - 9)/(x - 3).
Simplify (x² + 5x + 6)/(x² - 4).
Simplify (2x + 6)/(x² + 3x).
ADD AND SUBTRACT
  • Read the question.
  • Tap your answer.
What is (1)/(x) + (1)/(x + 1)?
What is (2)/(x - 1) + (3)/(x + 2)?
What is (x)/(x + 3) - (2)/(x + 3)?
ADD 1/X + 1/(X + 1), IN ORDER
  • ?Rewrite 1/(x + 1) as x/(x(x + 1))
  • ?Write the result: (2x + 1)/(x² + x)
  • ?Add the tops: (x + 1) + x = 2x + 1
  • ?Rewrite 1/x as (x + 1)/(x(x + 1))
  • ?Find the common denominator: x(x + 1)
WHY THIS EXERCISEAdding fractions with x follows the same four moves as adding 1/2 + 1/3. The letters do not change the rules.
StatementTrue or false?
(x² - 9)/(x - 3) simplifies to x + 3.?
In (x + 3)/x, you may cancel the x on top with the x on the bottom.?
1 ÷ 2 + 1 ÷ 3 = 5 ÷ 6?
The common denominator of 1/x and 1/(x + 1) is x(x + 1).?
(2 × 2 - 4) ÷ 2 × (3 × 2) ÷ (2 + 2) = 1?
WHY THIS EXERCISERational expressions obey fraction rules. A quick number check catches a canceling mistake every time.
Try it
Write (x² - 1)/(x + 1) on an index card. Factor the top, cancel, and write the result.
Put x = 3 into both the original and your answer. Do they match? Write what you notice.

Excellent. Tomorrow the measuring pitcher comes out in the Glass House Lab and your fractions get wet.

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