Polynomial Operations — Drills
Polynomial Operations has 7 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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).
Adding
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Simplify (2x2 + 7x) + (5x2 - 7x).
Worked answer
7x^2 + 0x
2x2 + 5x2 = 7x2
7x - 7x = 0x
Problem 2
Simplify (4x2 + 6x) + (1x2 + 8x).
Worked answer
5x^2 + 14x
4x2 + 1x2 = 5x2
6x + 8x = 14x
Subtracting
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Simplify (6x2 + 1x + 8) - (3x2 + 2x - 2).
Worked answer
3x^2 - 1x + 10
Signs flip: - 3x2 - 2x + 2
x2: 6 - 3 = 3
x: 1 - 2 = -1
Constants: 8 - (-2) = 10
Problem 2
Simplify (8x2 - 2x - 1) - (2x2 + 5x - 5).
Worked answer
6x^2 - 7x + 4
Signs flip: - 2x2 - 5x + 5
x2: 8 - 2 = 6
x: -2 - 5 = -7
Constants: -1 - (-5) = 4
Multiplying
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Expand (3x + 1)(5x - 1).
Worked answer
15x^2 + 2x - 1
3x · 5x = 15x2
Middle: -3x + 5x = 2x
1 × -1 = -1
Problem 2
Expand (4x - 2)(3x - 5).
Worked answer
12x^2 - 26x + 10
4x · 3x = 12x2
Middle: -20x - 6x = -26x
-2 × -5 = 10
Squares
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Expand (2x + 6)2.
Worked answer
4x^2 + 24x + 36
(2x + 6)(2x + 6)
4x2
Middle: twice 2x × 6, so 24x
(6)2 = 36
Problem 2
Expand (3x - 5)2.
Worked answer
9x^2 - 30x + 25
(3x - 5)(3x - 5)
9x2
Middle: twice 3x × -5, so -30x
(-5)2 = 25
Closure
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Polynomials are not closed under which operation?
- Subtraction
- Division
- Addition
- Multiplication
Worked answer
Division
x2 + 1x = x + 1x
That is not a polynomial.
Add, subtract and multiply always stay inside.
Degree
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
What is the degree of (x1 + 1)(x3 - 4)?
Worked answer
4
1 + 3 = 4
The leading term is x4.
Problem 2
What is the degree of (x2 + 1)(x2 - 4)?
Worked answer
4
2 + 2 = 4
The leading term is x4.
Area
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A rectangle has length x + 11 and width x - 9. Write its area as a polynomial.
Worked answer
x^2 + 2x - 99
(x + 11)(x - 9)
x2 - 9x + 11x - 99
= x2 + 2x - 99
Problem 2
A rectangle has length x + 2 and width x - 7. Write its area as a polynomial.
Worked answer
x^2 - 5x - 14
(x + 2)(x - 7)
x2 - 7x + 2x - 14
= x2 - 5x - 14