Adding is easy, subtracting is where the marks go, and multiplying is bookkeeping.
Let P = 3x² − 5x + 2 and Q = x² + 4x − 7
Find P − Q
Subtracting flips every sign in Q: 3x² − 5x + 2 − x² − 4x + 7 = 2x² − 9x + 9. D changes only the first term and adds the rest. Note −(−7) becomes +7, which is where the +9 comes from.
| Word | Means | In 3x² − 5x + 2 |
|---|---|---|
| term | a piece separated by + or − | 3x², −5x and 2 |
| coefficient | the number in front | 3 and −5 |
| degree | the highest exponent | 2 — a quadratic |
| constant | the term with no letter | 2 |
| standard form | powers running downhill | x², then x, then the number |
The sign in front belongs to the term. −5x is one object, not a 5x with a floating minus — writing it that way is what makes subtraction safe later.
Like terms have the same letter and the same exponent. x² and x are not alike.
3x² − 5x + 2
+ x² + 4x − 7
4x² − x − 5
3x² + x² = 4x²
x² has an unwritten 1
−5x + 4x = −x
still negative
2 − 7 = −5
the constants
3x² + x² is 4x², not 4x⁴. You are counting how many x² you have, the same way 3 apples and 1 apple make 4 apples — not 4 apple².
Change every sign in the second polynomial, then add as usual.
(3x² − 5x + 2) − (x² + 4x − 7)
as written
= 3x² − 5x + 2 − x² − 4x + 7
all three signs flip
= 2x² − 9x + 9
now combine
Flipping only the first term and copying the rest gives 2x² − x − 5, which is wrong twice over — and it is the most common single error in this topic. Rewrite the bracket with its new signs before you combine anything.
A number in front of the bracket multiplies every term first, sign included: 3(x² − x + 2) − 2(2x² + x − 5) = 3x² − 3x + 6 − 4x² − 2x + 10 = −x² − 5x + 16.
FOIL is only the two-by-two case. This is two by three, so expect six products.
| × | x² | −2x | +5 |
|---|---|---|---|
| x | x³ | −2x² | 5x |
| +3 | 3x² | −6x | 15 |
x³ − 2x² + 5x + 3x² − 6x + 15
all six, unsorted
−2x² + 3x² = x²
collect the squares
5x − 6x = −x
collect the x terms
x³ + x² − x + 15
standard form
x · x² = x³
1 + 2 = 3
x + x² stays x + x²
nothing to combine
| Operation | Degree of the answer |
|---|---|
| add or subtract | the higher of the two — or lower, if the tops cancel |
| multiply | the two degrees added: 1 + 2 = 3 |
A degree check is a fast filter on multiple choice. A linear times a quadratic must be cubic, so any choice starting with x⁴ or x² is gone before you calculate.
Subtract 3x − 5 from x² − 2.
(x² − 2) − (3x − 5)
“from” names the one that goes first
x² + 0x − 2 − 3x + 5
no x term in the first: write 0x
x² − 3x + 3
x = 2: 2 − 1 = 1, and 4 − 6 + 3 = 1 ✓
Different: the order is reversed from the reading order, and a gap in the powers needs a placeholder.
(3x − 5) − (x² − 2) = −x² + 3x − 3: every sign of the right answer flipped. It will be one of the choices.
Add (½x² − ⅓x + 2) + (¼x² + x − 5).
½ + ¼ = 2 + 14 = ¾
the x² terms
−⅓ + 1 = ⅔
the x terms; x means 1x
2 − 5 = −3, so ¾x² + ⅔x − 3
the constants
Different: each column needs its own common denominator; the letters do nothing new.
Simplify (x² + 3x − 1) − (x² − 2x + 4).
x² + 3x − 1 − x² + 2x − 4
flip all three signs
0x² + 5x − 5 = 5x − 5
x = 2: 9 − 4 = 5 ✓
Different: the squares cancel, so two quadratics have a linear difference — the degree can drop.
“What must be added to A to obtain B?” The missing piece is B − A. What must be added to 2x² + x − 4 to obtain 5x² − 3x + 1?
(5x² − 3x + 1) − (2x² + x − 4)
target minus start
= 5x² − 3x + 1 − 2x² − x + 4 = 3x² − 4x + 5
add back: 2x² + x − 4 + 3x² − 4x + 5 ✓
One coefficient only. Find the coefficient of x² in (2x² − x + 4)(x² + 3x − 2). Skip the full expansion — list only the pairs whose exponents add to 2.
(2x²)(−2) = −4x²
x² × constant
(−x)(3x) = −3x²
x × x
(4)(x²) = 4x²
constant × x²
−4 − 3 + 4 = −3
three pairs, not nine products
Identities: match coefficients. If (x + b)(2x − 1) = 2x² + 7x − 4 for all x, find b. “For all x” means the two sides are the same polynomial, term by term.
(x + b)(2x − 1) = 2x² + (2b − 1)x − b
expand with b as a number
2b − 1 = 7 → b = 4
match the x terms
−b = −4 ✓
the constants agree too
In the coefficient shortcut the middle pair (x times x) is the one people miss, giving 0 here instead of −3. Before adding, check that you have one pair for every term of the first bracket that can reach x².
Polynomial items are rarely a plain expand-everything. Expect a subtraction with a sign trap, “what must be added”, or a single coefficient. Read which one is asked, use the shortcut, and test the finished choice with x = 2 as in the check below.
These are identities, so any x works. Use x = 2 on the originals and on the answers.
Let A = 2x² + x − 6 and B = 5x² − 3x + 4
Find B − A
Flip all of A's signs: 5x² − 3x + 4 − 2x² − x + 6 = 3x² − 4x + 10. C adds instead of subtracting. The +10 comes from 4 − (−6), the term most often missed.
