Polynomial Operations — Challenge
Add and subtract by collecting like terms. Multiply by making sure every pair meets exactly once.
What these problems 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 (2x2+3x)+(x2-x).
- 3x2+4x
- 3x2+2x
- 2x2+2x
- 3x3+2x
- x2+2x
Worked answer
3x2+2x
Add the like terms separately. The x2 terms give 2x2+x2 = 3x2.
The x terms give 3x-x = 2x.
So the sum is 3x2+2x. Powers are not added when terms are combined, which rules out 3x3.
Problem 2
Expand (x+2)(x-5).
- x2+3x-10
- x2-3x+10
- x2-3x-10
- x2-7x-10
- x2-10
Worked answer
x2-3x-10
Multiply every term in the first bracket by every term in the second.
x· x = x2, x·(-5) = -5x, 2· x = 2x and 2·(-5) = -10.
Collecting the middle terms gives -5x+2x = -3x, so the answer is x2-3x-10.
Problem 3
What is the degree of 4x3-2x+7?
- 1
- 2
- 3
- 4
- 7
Worked answer
3
The degree is the highest power of the variable that appears.
The powers here are 3, 1 and 0, so the degree is 3.
The 4 is the leading coefficient, which is a different thing.
Problem 4
Simplify (3x2-x)-(x2+2x).
- 2x2+x
- 2x2-3x
- 2x2-x
- 4x2+x
- 2x2+3x
Worked answer
2x2-3x
The minus sign applies to everything in the second bracket, so it becomes -x2-2x.
The x2 terms give 3x2-x2 = 2x2.
The x terms give -x-2x = -3x, so the answer is 2x2-3x.
Problem 5
Expand 3x(x2-4).
- 3x3-4x
- 3x2-12x
- 3x3-12x
- 3x3-12
- x3-12x
Worked answer
3x3-12x
Multiply both terms inside the bracket by 3x.
3x· x2 = 3x3, since multiplying powers adds the exponents.
3x·(-4) = -12x, so the answer is 3x3-12x.
Problem 6
Expand (x+4)2.
- x2+16
- x2+8x+16
- x2+4x+16
- x2+8x+8
- x2+16x+16
Worked answer
x2+8x+16
Squaring a bracket means multiplying it by itself: (x+4)(x+4).
The terms are x2, 4x, 4x and 16.
The two middle terms add to 8x, giving x2+8x+16. Squaring each term separately would wrongly give x2+16.
Problem 7
What is the leading coefficient of -5x4+2x?
- -5
- 5
- 4
- 2
- -4
Worked answer
-5
The leading term is the one with the highest power, which is -5x4.
Its coefficient, including the sign, is -5.
The 4 is the degree, not the coefficient.
Problem 8
How many terms does 2x3-x2+7 have?
- 1
- 2
- 3
- 4
- 7
Worked answer
3
Terms are the parts separated by plus or minus signs.
Here they are 2x3, -x2 and 7, so there are three terms.
A three-term polynomial is called a trinomial.