7日間コース
Master Polynomials & Equations in 7 Days
From polynomial arithmetic to monomial multiplication and equation transformation — build the algebraic fluency needed for junior high 2 and beyond.
対象: Junior High 2 / Overseas Japanese students
Concepts & Worked Examples
Understand polynomial operations, monomial multiplication/division, using expressions for proofs, and transforming equations.
同類項（同じ文字の同じ次数の項）をまとめて計算する。
例: 5x + 3y - 2x + y = (5x - 2x) + (3y + y) = 3x + 4y
分配法則: a(b + c) = ab + ac
例: (2a + 3b) + (4a - 5b) = 2a + 3b + 4a - 5b = 6a - 2b
例: (x - 6y) - (3x + 2y) = x - 6y - 3x - 2y = -2x - 8y
式に数をかける: 4(2x - 5y) = 8x - 20y
式を数で割る: (6x + 9y) ÷ 3 = 2x + 3y
カッコや分数を含む式:
例: 3(2x + 4y) + 2(x - 9y) = 6x + 12y + 2x - 18y = 8x - 6y
例: 2(5a + 2y) - 4(2a - 3y) = 10a + 4y - 8a + 12y = 2a + 16y
多項式の分数: (a - 3b)/4 - (a - 5b)/20 最小公倍数 = 20: = 5(a - 3b)/20 - (a - 5b)/20 = (5a - 15b - a + 5b)/20 = (4a - 10b)/20 = (2a - 5b)/10
単項式 × 単項式: 係数同士をかけて、文字同士もかける。
7a × (-3y) = -21ay (-4a²) × (-6ab) = 24a³b
単項式 ÷ 単項式: 係数を割って、文字も割る。
16ay ÷ (-2a) = (16 ÷ -2) × (ay ÷ a) = -8y
累乗のかけ算: a² × a³ = a⁵（指数を足す）
文字式を使って数の性質を証明する。
証明の手順:
方程式を特定の文字について解く。
例: 4x - 8y = 5 を x について解く 4x = 5 + 8y x = (5 + 8y)/4
例: ℓ = 2πr を r について解く r = ℓ/(2π)
コツ: 求める文字以外はすべて「数」と思って扱う。
Simplify: (2a + 3b) + (4a - 5b)
解答
Remove brackets: = 2a + 3b + 4a - 5b Combine like terms: = (2a + 4a) + (3b - 5b) = 6a - 2b
When adding polynomials, simply combine like terms.
Calculate: 7a × (-3y)
解答
Multiply coefficients: 7 × (-3) = -21 Multiply variables: a × y = ay 7a × (-3y) = -21ay
Multiply the numerical parts and variable parts separately.
Solve a = (1/2)bc for b
解答
Multiply both sides by 2: 2a = bc Divide both sides by c: b = 2a/c
Isolate b by undoing the operations around it: multiply by 2, then divide by c.
Simplify: (a - 3b)/4 + (3a - 4b)/5
解答
LCD = 20 = 5(a - 3b)/20 + 4(3a - 4b)/20 = (5a - 15b + 12a - 16b)/20 = (17a - 31b)/20
Find the LCD, multiply each fraction accordingly, then combine numerators.
Basic Practice (★1-2)
10 problems covering Units 9-13. Practice polynomial operations and equation transformation.
単元9〜13の演習問題。途中式を丁寧に書こう。
★1 Simplify: 4a + 2b + 5a - 3b
解答
= (4a + 5a) + (2b - 3b) = 9a - b
Combine like terms: a-terms together, b-terms together.
★1 Calculate: 16xy ÷ (-2a) × 9a (assume typo fix: 16ay ÷ (-2a))
解答
16ay ÷ (-2a) = -8y
Divide coefficients: 16 ÷ (-2) = -8. Divide variables: ay ÷ a = y.
★2 Simplify: 3(x + 2y) - 2(x - 4y)
解答
= 3x + 6y - 2x + 8y = x + 14y
Distribute, then combine like terms. Watch the sign on -2(x - 4y).
★2 Solve 5x - y = 4 for y
解答
5x - y = 4 -y = 4 - 5x y = -4 + 5x y = 5x - 4
Move all non-y terms to the other side, then multiply by -1.
Standard Practice (★2-3)
10 problems at standard difficulty covering all polynomial and equation manipulation skills.
Pattern Drills (★2-3)
10 problems focusing on fractional expressions, proof setups, and multi-step equation solving.
Applied Problems (★3-4)
10 challenging problems combining polynomial arithmetic with equation transformation.
Advanced Challenge (★4-5)
10 high-difficulty problems including algebraic proofs and complex equation rearrangement.
Comprehensive Test & Bridge to Expansion
Mixed test covering ★1-5. Plus: how polynomial skills connect to expansion and factorization in junior high 3.