Polynomial Operations — Print
Add and subtract by collecting like terms. Multiply by making sure every pair meets exactly once.
What the practice questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Like terms — Terms with exactly the same variable part; only these can be combined.
- Area — Space inside, measured in square units.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Distribute — Multiply a bracket out: 3(x + 4) = 3x + 12.
- Polynomial — A sum of terms, each a number times a whole-number power of the variable.
- Difference of squares — a2 - b2 = (a - b)(a + b).
Draw a rectangle x + 3 wide and x + 2 tall. What is its area?
Its area is (x + 3)(x + 2) — that is just what area means. But you can also cut it up.
The two cuts make four patches, and each patch is easy: x · x, 3 · x, 2 · x and 3 · 2.
Add them: x2 + 3x + 2x + 6 = x2 + 5x + 6.
Nothing can be missed, because no part of a rectangle is left uncovered.
That is all FOIL ever meant — it just named the four patches in an order. The picture keeps working for bigger brackets, where FOIL does not: a 3 × 2 grid has six patches, and you fill in all six.
Adding and subtracting
Add by collecting like terms. (3x2 + 5x) + (2x2 - 8x) = 5x2 - 3x.
Subtract by distributing the minus sign to every term in the second polynomial, then collecting.
(4x2 - 3x + 1) - (x2 + 5x - 6) = 4x2 - 3x + 1 - x2 - 5x + 6 = 3x2 - 8x + 7.
The minus reaches everything
Subtracting (x2 + 5x - 6) gives -x2 - 5x + 6. All three signs change, including the one that was already negative.
Multiplying
Every term in the first bracket must multiply every term in the second. For two binomials that is four products — the origin of FOIL.
For bigger brackets, FOIL does not apply, but the principle does. Use a grid and fill in every cell.
Then collect like terms and write the answer in standard form.
Use a box, not a mnemonic
Draw a rectangle with one polynomial along the top and the other down the side. Fill every cell, then add. It never misses a product, whatever the sizes.
Special products worth knowing
(a + b)2 = a2 + 2ab + b2. The middle term is easy to forget.
(a - b)2 = a2 - 2ab + b2.
(a + b)(a - b) = a2 - b2 — the middle terms cancel, leaving a difference of squares.
(x + 3)2 is not x2 + 9
It is (x+3)(x+3) = x2 + 6x + 9. Write the bracket out twice. Squaring a sum is not squaring each piece.
Closure
Add, subtract or multiply two polynomials and you always get a polynomial. That is what closure means.
Division breaks it: x2 + 1x is not a polynomial, because of the 1x.
So polynomials are closed under addition, subtraction and multiplication, but not division — exactly like the integers.
Worked examples
Subtracting
Simplify (5x2 - 2x + 7) - (2x2 + 4x - 3).
- Distribute the minus: 5x2 - 2x + 7 - 2x2 - 4x + 3 Every sign in the second bracket flips.
- x2 terms: 5x2 - 2x2 = 3x2 Collect like terms.
- x terms: -2x - 4x = -6x Both negative now.
- Constants: 7 + 3 = 10 -(-3) became +3.
3x2 - 6x + 10
Multiplying binomials
Expand (2x - 5)(3x + 4).
- 2x · 3x = 6x2 First times first.
- 2x · 4 = 8x and -5 · 3x = -15x The two middle products.
- -5 · 4 = -20 Last times last.
- 6x2 + 8x - 15x - 20 = 6x2 - 7x - 20 Collect the middle terms.
6x2 - 7x - 20
A special product
Expand (3x - 4)2.
- Write it out: (3x - 4)(3x - 4) Never square the terms separately.
- 9x2 from the first pair. 3x · 3x.
- Middle: -12x - 12x = -24x Twice the product of the two terms.
- Last: 16 (-4)(-4) = +16.
9x2 - 24x + 16
Words to know
- Polynomial
- A sum of terms, each a number times a whole-number power of the variable.
- Like terms
- Terms with identical variable parts. 4x2 and -9x2 are like; 4x2 and 4x are not.
- Binomial
- A polynomial with two terms.
- Closure
- Polynomials added, subtracted or multiplied always give another polynomial.
- Standard form
- Terms written in order of decreasing degree.
Mistakes to watch for
Distributing a subtraction to only the first term. -(x2 + 5x - 6) is -x2 - 5x + 6: all three signs change, including the one that was already negative. This is the most reliable source of errors in the unit.
Writing (x + 5)2 as x2 + 25. Squaring a sum is not squaring each piece — write the bracket out twice and the middle term 10x appears. Check with x = 1: 36, not 26.
Adding 3x2 and 4x as like terms. They are not: x2 and x are different quantities, the way an area and a length are. Only identical variable parts can be collected.
Missing a product when the brackets get bigger. FOIL only has four letters and stops working past two binomials. Draw the grid and fill every cell — it never misses.
Algebra tiles — physical squares and rectangles for x2, x and 1 — became a standard classroom tool in the 1960s.
But the idea is far older: Euclid's Book II proves algebraic identities entirely with rectangles, because the Greeks had no algebraic notation.
His Proposition 4 is (a+b)2 = a2 + 2ab + b2, stated and proved as a fact about squares on a line.
Problem 1
Simplify (3x + 5) + (2x - 8).
Worked answer
5x - 3
3x + 2x = 5x
5 - 8 = -3
Problem 2
Simplify (x2 + 4x) + (3x2 - x).
Worked answer
4x^2 + 3x
x2 + 3x2 = 4x2
4x - x = 3x
Problem 3
Simplify (5x - 2) - (2x + 7).
Worked answer
3x - 9
5x - 2 - 2x - 7
= 3x - 9
Problem 4
Expand 3x(x + 4).
Worked answer
3x^2 + 12x
3x · x = 3x2
3x · 4 = 12x
Problem 5
Expand (x + 3)(x + 5).
Worked answer
x^2 + 8x + 15
x2
5x + 3x = 8x
15
Problem 6
Simplify (4x2 - 3x + 6) - (x2 + 5x - 2).
Worked answer
3x^2 - 8x + 8
4x2 - x2 = 3x2
-3x - 5x = -8x
6 + 2 = 8
Problem 7
Expand (2x - 3)(x + 4).
Worked answer
2x^2 + 5x - 12
2x2
8x - 3x = 5x
-12
Problem 8
Expand (3x - 2)(3x + 2).
Worked answer
9x^2 - 4
9x2
6x - 6x = 0
-4
A difference of squares
Problem 9
Expand (x + 6)2.
Worked answer
x^2 + 12x + 36
(x+6)(x+6)
x2 + 6x + 6x + 36
= x2 + 12x + 36
Problem 10
Expand (2x - 5)2.
Worked answer
4x^2 - 20x + 25
4x2
-10x - 10x = -20x
+25
Problem 11
Polynomials are not closed under which operation?
- Division
- Addition
- Subtraction
- Multiplication
Worked answer
Division
x2 + 1x = x + 1x
That is not a polynomial
Problem 12
What is the degree of (x2 + 1)(x3 - 4)?
Worked answer
5
2 + 3 = 5
The leading term is x5
Problem 13
Expand (x + 2)(x2 - 3x + 5).
Worked answer
x^3 - x^2 - x + 10
x3 - 3x2 + 5x
+ 2x2 - 6x + 10
= x3 - x2 - x + 10
Problem 14
A rectangle has length x + 7 and width x - 3. Write its area as a polynomial.
Worked answer
x^2 + 4x - 21
(x + 7)(x - 3)
x2 - 3x + 7x - 21
= x2 + 4x - 21
Problem 15
If A = 3x2 - x + 4 and B = x2 + 2x - 5, find A - 2B.
Worked answer
x^2 - 5x + 14
2B = 2x2 + 4x - 10
3x2 - x + 4 - 2x2 - 4x + 10
= x2 - 5x + 14
Problem 16
Expand and simplify (x + 4)2 - (x - 4)2.
Worked answer
16x
(x2 + 8x + 16) - (x2 - 8x + 16)
= 16x
The squares cancel entirely