Week 08 · Tuesday: Polynomials (Algebra 1 9-12)

Tuesday: Add, Subtract, Multiply

Wren pins two pad plans to the wall. "Plan A is x² + 5x + 6. Plan B is 2x² + 3x + 1. We might build both."

"Then add them," Comet says. "Squares with squares, strips with strips, units with units."

"Like terms," Wren says. "x² + 5x + 6 plus 2x² + 3x + 1 is 3x² + 8x + 7. What do you notice about the answer?"

"It is still a polynomial," Comet says. "Add two polynomials, get a polynomial. Like adding whole numbers."

Nova projects a product, (x + 1)(2x + 3). "Would you like a hint? Every term in the first meets every term in the second."

"2x², then 3x, then 2x, then 3," Comet says. "2x² + 5x + 3. Still a polynomial."

"Our engineer," Wren says, "subtract plan A from plan B, and tell us how many more tiles B needs."

Adding and subtracting

Line up like terms: x² with x², x with x, numbers with numbers.
To add, add the coefficients of each pair of like terms.
To subtract, subtract every term of the second polynomial, then combine like terms.
Write the answer from the highest power down.

Plan A plus plan B: (x² + 5x + 6) + (2x² + 3x + 1) = 3x² + 8x + 7.

Plan B minus plan A: (2x² + 3x + 1) - (x² + 5x + 6) = x² - 2x - 5. Every term of A is subtracted, including the 6.

A check with a number: at x = 2, A is 20 and B is 15. Their sum is 2015, and 3x² + 8x + 7 at x = 2 is 35.

Multiplying, two ways

Way one is the area model. Lay tiles for (x + 1)(2x + 3) and count: two x² squares, five strips, three units.

Way two is distributing. Every term of the first polynomial multiplies every term of the second, then like terms combine.

Both ways give the same polynomial. The tiles show why distributing works; distributing works even when tiles would be awkward.

Multiply	x	+ 1
2x	2x²	2x
+ 3	3x	3

Adding the four boxes: 2x² + 3x + 2x + 3 = 2x² + 5x + 3.

Closed, like the whole numbers

Add, subtract or multiply two whole numbers and you get a whole number. The whole numbers are closed under those operations.

Polynomials work the same way. Add, subtract or multiply two polynomials and the answer is always a polynomial.

Division is different for both. 7 ÷ 2 is not a whole number, and x ÷ (x + 1) is not a polynomial.

ADD AND SUBTRACT

Read the question.
Tap your answer.

What is (x² + 5x + 6) + (2x² + 3x + 1)?

What is (2x² + 3x + 1) - (x² + 5x + 6)?

What is (x² - 4x + 7) + (3x² + 4x - 2)?

What is (5x² + 2x) - (x² + 2x + 9)?

MULTIPLY (X + 1)(2X + 3) BY DISTRIBUTING

?Multiply 1 by 3: 3.
?Multiply 1 by 2x: 2x.
?Multiply x by 3: 3x.
?Multiply x by 2x: 2x².
?Combine like terms: 2x² + 5x + 3.

WHY THIS EXERCISEDistributing in a fixed order means no product gets skipped, which is the most common slip.

MULTIPLY

Read the question.
Tap your answer.

What is (x + 1)(2x + 3)?

What is (x + 4)(x - 2)?

What is (2x + 1)(x + 5)?

Which operation can take you outside the polynomials, the way 7 ÷ 2 leaves the whole numbers?

Statement	True or false?
(x² + 5x + 6) + (2x² + 3x + 1) = 3x² + 8x + 7	?
(2x² + 3x + 1) - (x² + 5x + 6) = x² - 2x - 5	?
(x + 4)(x - 2) = x² + 2x - 8	?
(x + 4)(x - 2) = x² - 8	?
The degree of (x + 1)(2x + 3) is 2.	?
Adding or multiplying two polynomials always gives a polynomial.	?
Dividing one polynomial by another always gives a polynomial.	?

WHY THIS EXERCISEEvery answer here is a polynomial. That is closure, and it is why polynomial algebra always has somewhere to go.

Try it

Write two polynomials of your own with degree 2. Add them, subtract them and multiply them.
Check one answer with a number: pick x = 2, work out both sides, and see that they agree.

Loft Log: write the steps for subtracting polynomials and the slip you most want to avoid.Explain why the product of two degree 2 polynomials has degree 4. Use the tiles or the boxes.Write one way polynomials are like whole numbers and one way they are different.

Excellent work. Tomorrow you cut your own tiles in the Loft Lab and build rectangles with them.