Equations and Proofs

Introduction: Welcome to the Adventure of Mathematics II!

Hello everyone! Today marks the beginning of our journey into "Mathematics II." The first chapter, "Expressions and Proofs," is a vital unit that helps us organize the "toolbox" we’ll need to solve future math problems.
You might think, "This looks like a lot of difficult calculations..." but don't worry. Once you master how to use each tool (formula), you'll start to see that it’s as fun as solving a puzzle. Let's start by getting comfortable with these new rules of calculation!

1. Expansion and Factorization of Cubic Expressions

In Mathematics I, we dealt with quadratic expressions (such as \(x^2\)), but in Mathematics II, we introduce cubic expressions (such as \(x^3\)).

【Expansion Formulas】

The magic rules for removing parentheses:
① \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
② \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)
(Tip: The signs in ② alternate as "plus, minus, plus, minus"!)

【Factorization Formulas】

Turning separate terms into a parenthetical form:
③ \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
④ \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

【Common Mistake!】
In the factorization formula, the part inside the second set of parentheses, \((a^2 - ab + b^2)\), has \(ab\), not \(2ab\). Be careful not to confuse this with the quadratic expansion formula \((a+b)^2 = a^2 + 2ab + b^2\).

★Tip:
At first, it’s perfectly fine to write these formulas on a piece of paper and refer to them while solving problems. Through repetition, your hands will naturally learn them!

2. The Binomial Theorem

This is a powerful rule for expanding expressions like \((a+b)^n\), where the binomial is raised to a high power.

Pascal's Triangle and Coefficients

When you expand \((a+b)^1, (a+b)^2, (a+b)^3 \dots\), the coefficients follow an interesting pattern. Arranging these in a pyramid gives us "Pascal's Triangle."

【The Binomial Theorem Formula】
\((a+b)^n = {}_n C_0 a^n + {}_n C_1 a^{n-1}b + {}_n C_2 a^{n-2}b^2 + \dots + {}_n C_n b^n\)
It may look intimidating, but the rules are simple:
1. The coefficients use combinations: \({}_n C_r\).
2. The power of \(a\) decreases by 1 with each term.
3. The power of \(b\) increases by 1 with each term.

💡Fun Fact:
If you're asked, "What is the coefficient of \(x^7y^3\) in the expansion of \((x+y)^{10}\)?" you don't need to expand the whole thing! Using the Binomial Theorem, you can find the answer instantly by calculating \({}_{10} C_3\).

3. Division of Polynomials

Just like numerical division, you can also divide polynomials.

The Basics of Polynomial Division

When you divide polynomial \(A\) by polynomial \(B\), resulting in a quotient \(Q\) and a remainder \(R\), the following relationship holds:
\(A = B \times Q + R\)
(Note: The degree of the remainder \(R\) must always be lower than the degree of the divisor \(B\).)

【Steps: Tips for Long Division】
1. Focus on the highest-degree term and think about what you need to multiply it by to eliminate it.
2. Subtract, then "bring down" the next terms one by one.
3. You're finished once the degree of the remainder is lower than the divisor!

4. Fractional Expressions and Their Calculations

Expressions containing variables (like \(x\)) in the denominator are called fractional expressions. The way to handle them is exactly the same as the fractions you learned in elementary school!

・Simplification: Divide the numerator and denominator by the same expression.
・Addition/Subtraction: Use "common denominators" to combine them.
・Multiplication/Division: Multiply numerators with numerators and denominators with denominators (for division, multiply by the reciprocal).

【Tip: Strategy for Complex Fractions】
Before calculating, try to factor the numerator and denominator first. You'll often find common terms that make the calculation much easier.

5. Identities

Do you know the difference between an "equation" and an "identity"? This is an important distinction!

・Equation: \(x + 1 = 3\) (Only true when \(x=2\). Used to find a specific solution.)
・Identity: \((x+1)^2 = x^2 + 2x + 1\) (True for any value of \(x\).)

How to Solve Identities (Determining Coefficients)

There are two ways to find unknown coefficients (like \(a, b, c\)):
1. Method of Undetermined Coefficients: Expand both sides and compare the coefficients of corresponding terms.
2. Substitution Method: Substitute easy values (like 0 or 1) into \(x\) to create equations.

6. Proofs of Equalities and Inequalities

Here is the strategy to use when you are asked to "prove that this expression is true."

【Proof of Equalities】

The basic approach is to "calculate the (left-hand side) to match the form of the (right-hand side)."
If that's difficult, show that "(left-hand side) - (right-hand side) = 0."

【Proof of Inequalities】

To prove \(A > B\), subtract them and show that \(A - B > 0\).
Your go-to "weapon" here is:
(Real number)\(^2 \geqq 0\) (Anything squared is always 0 or greater!)

【Super Important: Arithmetic Mean and Geometric Mean】

For \(a > 0, b > 0\), the following relationship holds:
\(\frac{a+b}{2} \geqq \sqrt{ab}\)
(Memory tip: It is often used in the form \(a+b \geqq 2\sqrt{ab}\)!)

💡Analogy:
The sum (arithmetic mean) is always greater than or equal to the square root of the product (geometric mean). This is the ultimate tool for solving "find the minimum value" problems.

【Summary: The Essence of Proofs】
Since the "goal" (like the form of the right-hand side) is visible in proof problems, you’re less likely to get lost. Treat it like a puzzle—enjoy figuring out how to manipulate the expressions to reach the finish line.

Final Thoughts

How did you find the "Expressions and Proofs" chapter?
It might feel like a lot of formulas at first, but once you internalize them, all of Mathematics II will become much easier. "Expansion formulas" and the "Arithmetic-Geometric Mean Inequality" are like "sidekicks" you'll encounter again and again.
Don't rush; take it one step at a time. I'm rooting for you!