Polynomial Operations — Practice
Add and subtract by collecting like terms. Multiply by making sure every pair meets exactly once.
What these 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).
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