A2.A.1 — Add and subtract polynomials.
Simplify
(2x3−4x2+5)−(3x3+x2−7)
Step 01 of 04
Same skill as Algebra I, but degrees are higher and you need to track more like-term groups. Same rule: combine terms with identical variable parts.
Step 02 of 04
Distribute the minus sign.
=2x3−4x2+5−3x3−x2+7
Step 03 of 04
Group and combine.
=(2x3−3x3)+(−4x2−x2)+(5+7)
=−x3−5x2+12
Step 04 of 04
Standard form. Always write the result with terms in DESCENDING power order. Missing degrees are simply absent (or you can write them with a zero coefficient as placeholders for division — see skill A2.A.3).
Key insight
Add / subtract polynomials = combine like terms after distributing any leading minus. Result lists terms in descending power. The mechanics don't change with degree — only the bookkeeping gets longer.
A2.A.2 — Multiply polynomials (including binomial powers).
Step 01 of 04
Every term times every term. FOIL is the binomial-times-binomial shortcut; for trinomials and bigger, use the general distributive rule.
(x+3)(x2−2x+5)
=x⋅x2+x(−2x)+x(5)+3x2+3(−2x)+3(5)
=x3−2x2+5x+3x2−6x+15=x3+x2−x+15
Step 02 of 04
Binomial powers — the binomial theorem (low orders worth memorizing).
| (a+b)2 | →a2+2ab+b2 |
| (a+b)3 | →a3+3a2b+3ab2+b3 |
| (a+b)4 | →a4+4a3b+6a2b2+4ab3+b4 |
The coefficients (1, 1) → (1, 2, 1) → (1, 3, 3, 1) → (1, 4, 6, 4, 1) formPascal's triangle. Each entry is the sum of the two above it.
Step 03 of 04
Worked example. Expand (2x−1)3. Use (a+b)3=a3+3a2b+3ab2+b3 with a=2x, b=−1:
(2x)3+3(2x)2(−1)+3(2x)(−1)2+(−1)3
=8x3−12x2+6x−1
Step 04 of 04
Watch for sign and coefficient. When b is negative, the signs alternate. When either has a coefficient, you must apply the power to the WHOLE term (e.g., (2x)2=4x2, not 2x2).
Key insight
Polynomial multiplication = every term times every term, then combine. For (a+b)n, the coefficients come from row n of Pascal's triangle; the powers of a decrease as the powers of b increase.
A2.A.3 — Polynomial long division.
Step 01 of 04
Polynomial long division is a generalization of regular long division — same "divide, multiply, subtract, bring down" rhythm, just with variable terms. It works for any divisor; synthetic division (next skill) is a shortcut that ONLY works for divisors of the form x−c.
Step 02 of 04
Setup. Write the dividend with placeholders for any missing degrees. Here, no missing degrees — write as is.
Step 1 — divide leading terms.
x2x3=2x2(first quotient term)
Multiply the divisor and subtract.
(2x2)(x−2)=2x3−4x2
(2x3−5x2)−(2x3−4x2)=−x2
Bring down the next term: −x2+3x.
Step 03 of 04
Step 2 — repeat with the new leading term.
x−x2=−x(next quotient term)
(−x)(x−2)=−x2+2x
(−x2+3x)−(−x2+2x)=x
Bring down: x−7.
Step 04 of 04
Step 3 — repeat once more.
xx=1(quotient term)
(1)(x−2)=x−2
(x−7)−(x−2)=−5(remainder)
The remainder is a constant — divisor was degree 1, so remainder is degree 0 or empty. Final answer:
2x2−x+1+x−2−5
Key insight
Same rhythm as numerical long division: divide leading terms → multiply → subtract → bring down → repeat. Stop when the running remainder has lower degree than the divisor. Always include placeholders for missing degrees.
A2.A.4 — Synthetic division.
Same dividend
x−22x3−5x2+3x−7
Step 01 of 04
Synthetic division is a streamlined version of long division that ONLY works for divisors of the form x−c (linear, monic). Same answer, much less writing.
For x−c, use +c as the divisor. Sign trap: dividing by x−2, use +2. By x+3, use −3.
Step 02 of 04
Step 1 — write coefficients only. Include zeros for missing degrees.
2x3−5x2+3x−7→[2,−5,3,−7]
Divisor x−2 → use 2.
Step 03 of 04
Step 2 — bring down, multiply, add. Bring down the first coefficient (2). Multiply by 2, add to the next coefficient. Repeat.
222−54−13−21−72−5
Bottom row: 2,−1,1,−5. The last entry is the remainder; the others are quotient coefficients in descending degree.
Step 04 of 04
Step 3 — read off the quotient.
quotient=2x2−x+1,remainder=−5
Same answer as long division. The quotient drops one degree from the dividend (since divisor is degree 1). For higher-degree divisors, synthetic division does NOT work — you must use long division.
Key insight
Synthetic division: write coefficients (with zero placeholders), use +c for divisor x−c, bring-down-multiply-add. Last bottom entry = remainder; the rest = quotient coefficients, degree-1 lower than the dividend.