Adding needs like terms. Multiplying needs every term to meet every other term.
Expand
(2x − 3)(x + 5)
FOIL: first 2x · x = 2x², outer 2x · 5 = 10x, inner −3 · x = −3x, last −3 · 5 = −15. Combining the middle terms gives 10x − 3x = 7x, so 2x² + 7x − 15. A multiplies only the firsts and the lasts and drops the middle entirely. B adds 10x and 3x, ignoring the minus on the 3.
| Polynomial | Terms | Name | Degree |
|---|---|---|---|
| 7x³ | 1 | monomial | 3 |
| x² − 4 | 2 | binomial | 2 |
| 2x² + 5x − 1 | 3 | trinomial | 2 |
| 3x²y⁴ | 1 | monomial | 6 |
The degree is the highest total exponent in any single term. In 3x²y⁴ the exponents add to 6, so the degree is 6 — not 4.
Subtracting means changing the sign of every term inside.
(4x² + 3x − 7) − (x² − 5x + 2)
note the minus
= 4x² + 3x − 7 − x² + 5x − 2
all three signs flipped
= 3x² + 8x − 9
combine
Writing 4x² + 3x − 7 − x² − 5x + 2 flips only one sign and is the error this topic is built around. The minus belongs to the whole bracket.
3x²(2x³ − 5x + 4)
three products
= 6x⁵ − 15x³ + 12x²
coefficients multiply, exponents add
3x² · 2x³ = 6x⁵ because 3 × 2 = 6 and x² · x³ = x⁵. The two parts are handled separately, every time.
The grid is FOIL without the acronym, and it extends: a trinomial times a binomial is a 3 × 2 grid with six boxes. Nothing new to learn.
The two diagonal boxes — the outer and inner products — are the ones that combine. If they do not share a letter part, there is no middle term to collect.
Every term on top gets divided, not just the first.
(12x⁵ − 8x³ + 4x²) ÷ 4x²
three separate divisions
= 12x⁵4x² − 8x³4x² + 4x²4x²
split it up
= 3x³ − 2x + 1
exponents subtract
A trinomial times a binomial is six products; a perimeter adds every side.
Multiply (x + 2)(3x² − x + 5).
(x + 2)(3x² − x + 5)
3 terms × 2 terms = 6 products
= 3x³ − x² + 5x + 6x² − 2x + 10
x times the bracket, then 2 times it
= 3x³ + 5x² + 3x + 10
−x² + 6x² = 5x², 5x − 2x = 3x
Check with x = 1: the question gives 3 × 7 = 21, and the answer gives 3 + 5 + 3 + 10 = 21 ✓.
A rectangle is (3x − 1) long and (x + 4) wide. Find its perimeter.
2(3x − 1) + 2(x + 4)
two lengths and two widths
= 6x − 2 + 2x + 8
distribute
= 8x + 6
combine like terms
Adding the length and the width once gives (3x − 1) + (x + 4) = 4x + 3, only half the way round. Multiplying them gives 3x² + 11x − 4, which is the area, not the perimeter.
The same moves; these are the three places students stall.
A difference that loses its x²: (2x² − 5x + 3) − (2x² + x − 4)
= 2x² − 5x + 3 − 2x² − x + 4
flip all three signs
= −6x + 7
the x² terms cancel
A quadratic minus a quadratic can come out linear. Do not write 0x² — just drop it.
A fraction coefficient: (x2 + 3)(4x − 2)
= 2x² − x + 12x − 6
x2 · 4x = 2x², x2 · (−2) = −x
= 2x² + 11x − 6
combine the middle
The grid does not care about fractions: multiply each pair, then collect.
A negative divisor: (−18x⁴ + 6x³ − 3x) ÷ (−3x)
= −18x⁴−3x + 6x³−3x + −3x−3x
split into three
= 6x³ − 2x² + 1
every sign flips
The last term is +1, because −3x divided by itself is 1. Multiply back: −3x(6x³ − 2x² + 1) = −18x⁴ + 6x³ − 3x ✓.
Periodical tests ask these straight — add, subtract, expand, divide — and often dress them as the perimeter or area of a rectangle. The fastest route is one sign-flip line for subtraction and the grid for products. For a 10-second check, put x = 1 into the question and into your answer: both must give the same number.
Simplify
(5x² − 2x + 6) − (3x² + 4x − 1)
Flip all three signs in the second bracket: 5x² − 2x + 6 − 3x² − 4x + 1. Then combine: 2x² − 6x + 7. C flips the first two signs but not the last, leaving 6 − 1 instead of 6 + 1. D adds instead of subtracting.
Hint: rewrite the subtraction as an addition with every sign in the second bracket reversed, before collecting anything.
Every lesson in the Grade 7-12 Math Review has one of these. 30 questions, marked as you answer, with the working shown for each.
Question 1 of 30
Add (3x + 5) + (2x − 7).
Combine like terms: 3x + 2x = 5x and 5 − 7 = −2.
Question 2 of 30
What is the degree of 4x³ − 2x + 7?
The degree is the highest exponent on the variable, which is 3.
Question 3 of 30
Multiply 3x(2x + 4).
Distribute the 3x to both terms: 3x × 2x = 6x² and 3x × 4 = 12x.
Question 4 of 30
Subtract (5x² − 3x) − (2x² + x).
The minus sign changes both terms in the second bracket: 5x² − 2x² = 3x² and −3x − x = −4x.
Question 5 of 30
Multiply (x + 3)(x + 5).
Multiply every term by every term: x² + 5x + 3x + 15 = x² + 8x + 15.
Question 6 of 30
Divide 12x⁵ by 3x².
Divide the numbers and subtract the exponents: 12 ÷ 3 = 4 and 5 − 2 = 3.
Question 7 of 30
Multiply (2x − 3)(x² + x − 4).
2x times the bracket gives 2x³ + 2x² − 8x, and −3 times it gives −3x² − 3x + 12. Adding: 2x³ − x² − 11x + 12.
Question 8 of 30
Simplify (4x² + 6x) ÷ 2x.
Divide EACH term by 2x: 4x² ÷ 2x = 2x and 6x ÷ 2x = 3. Dividing only the first term is the usual slip.
Question 9 of 30
A rectangle has length (2x + 5) and width (x − 3). What is its perimeter?
Perimeter is twice the length plus twice the width: 2(2x + 5) + 2(x − 3) = 4x + 10 + 2x − 6 = 6x + 4.
Question 10 of 30
Subtract (7x³ − 4x + 2) − (3x³ − 4x − 5).
Change every sign in the second bracket: 7x³ − 3x³ = 4x³, then −4x + 4x = 0, and 2 + 5 = 7.
Question 11 of 30
Add (4x − 9) + (6x + 2).
Like terms: 4x + 6x = 10x and −9 + 2 = −7, so 10x − 7. Answering 10x − 11 subtracts the 2 instead of adding it.
Question 12 of 30
By its number of terms, what is 3x⁴ − x + 8 called?
It has three terms, 3x⁴, −x and 8, so it is a trinomial. The exponent 4 is its degree, not its number of terms.
Question 13 of 30
Simplify −2x(x − 6).
−2x · x = −2x² and −2x · (−6) = +12x. Answering −2x² − 12x forgets that a negative times a negative is positive.
Question 14 of 30
Divide 18x⁶ by 6x³.
Coefficients divide, 18 ÷ 6 = 3, and exponents subtract, 6 − 3 = 3: 3x³. Answering 3x² divides the exponents; 3x⁹ adds them.
Question 15 of 30
Multiply (x + 2)(x + 9).
First x², outer 9x, inner 2x, last 18: x² + 11x + 18. Answering x² + 18 multiplies only the firsts and lasts.
Question 16 of 30
Subtract (8x² + 3x) − (5x² − 2x).
Flip both signs in the second bracket: 8x² + 3x − 5x² + 2x = 3x² + 5x. Answering 3x² + x leaves −2x unflipped.
Question 17 of 30
Simplify 2x²(4x³ − 3x² + x).
Coefficients multiply and exponents add: 8x⁵, −6x⁴, 2x³. Answering 8x⁶ − 6x⁴ + 2x² multiplies the exponents; 8x⁵ − 3x² + x distributes to the first term only.
Question 18 of 30
Multiply (x − 8)(x − 2).
Outer −2x, inner −8x, so the middle is −10x; last (−8)(−2) = +16: x² − 10x + 16. Answering x² − 10x − 16 gives the last term the wrong sign.
Question 19 of 30
Simplify (9x⁴ − 6x³ + 3x) ÷ 3x.
Divide each term: 3x³, −2x², and 3x ÷ 3x = 1. Answering 3x³ − 2x² treats the last term as 0; 3x⁴ − 2x³ + x divides only the coefficients.
Question 20 of 30
Simplify (3x² + 2x − 5) − (3x² − 4x + 1).
Flip all three signs: 3x² + 2x − 5 − 3x² + 4x − 1. The x² terms cancel, leaving 6x − 6. Answering −2x − 4 flips no signs at all; 6x − 4 flips all but the last.
Question 21 of 30
Multiply (2x + 7)(2x − 7).
Outer −14x and inner +14x cancel, leaving 4x² − 49. Answering 4x² + 28x − 49 adds the two middle terms instead of letting them cancel.
Question 22 of 30
A square has sides of (3x − 2). What is its perimeter?
Four equal sides: 4(3x − 2) = 12x − 8. Answering 9x² − 12x + 4 is the area; 12x − 2 multiplies only the first term by 4.
Question 23 of 30
Multiply (x + 4)(2x² − 3x + 1).
x times the bracket: 2x³ − 3x² + x. 4 times it: 8x² − 12x + 4. Combine: 2x³ + 5x² − 11x + 4. Answering 2x³ + 5x² − 13x + 4 gets the sign of the x term wrong (x − 12x is −11x).
Question 24 of 30
Expand (x/3 + 2)(6x − 9).
x/3 · 6x = 2x², x/3 · (−9) = −3x, 2 · 6x = 12x, 2 · (−9) = −18. Middle: 12x − 3x = 9x, so 2x² + 9x − 18. Answering 2x² + 15x − 18 adds 3x instead of subtracting it.
Question 25 of 30
Simplify (−24x⁵ + 16x³ − 4x²) ÷ (−4x²).
Dividing by a negative flips every sign: −24x⁵ ÷ (−4x²) = 6x³, 16x³ ÷ (−4x²) = −4x, −4x² ÷ (−4x²) = +1. Answering −6x³ + 4x − 1 divides by +4x² instead.
Question 26 of 30
A rectangle is (3x + 2) long and (x − 5) wide. What is its area?
(3x + 2)(x − 5) = 3x² − 15x + 2x − 10 = 3x² − 13x − 10. Answering 8x − 6 is the perimeter; 3x² + 17x − 10 adds 15x instead of subtracting it.
Question 27 of 30
Simplify (2x² − 3x + 4) − (2x² + 5x − 1).
Flip all three signs: 2x² − 3x + 4 − 2x² − 5x + 1 = −8x + 5. The x² terms cancel, so do not write 0x². Answering −8x + 3 leaves the −1 unflipped; 4x² − 8x + 5 adds the x² terms.
Question 28 of 30
Which product has NO x term when it is expanded?
In (3x − 5)(3x + 5) the outer +15x and the inner −15x cancel, giving 9x² − 25. (3x + 5)(3x + 5) has +15x + 15x = 30x, so its middle does not vanish.
Question 29 of 30
Multiply (x + 1)(x² − x + 1).
x times the bracket: x³ − x² + x. 1 times it: x² − x + 1. The x² and x terms cancel, leaving x³ + 1. Check with x = 1: (2)(1) = 2 and 1 + 1 = 2 ✓. Answering x³ + 2x² + 2x + 1 treats −x as +x.
Question 30 of 30
What must be added to 3x² − 5x + 2 to get 5x² + x − 4?
Subtract: (5x² + x − 4) − (3x² − 5x + 2) = 5x² + x − 4 − 3x² + 5x − 2 = 2x² + 6x − 6. Answering 2x² − 4x − 2 does not flip the signs of the second bracket; 8x² − 4x − 2 adds the two.
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