Abbreviated multiplication formulas and factoring methods are powerful tools that simplify computations and the solving of equations. On the YÖS exam these skills come up constantly. The ability to apply these formulas quickly and correctly can significantly speed up problem solving.
---
$$(a + b)^2 = a^2 + 2ab + b^2$$
In words: The square of the sum of two expressions equals the square of the first expression, plus twice the product of the first and the second, plus the square of the second expression.
Proof:$$(a + b)^2 = (a + b)(a + b) = a \cdot a + a \cdot b + b \cdot a + b \cdot b = a^2 + 2ab + b^2$$
Examples:$$(3 + 5)^2 = 9 + 30 + 25 = 64$$
check: $8^2 = 64$ ✓
$$(x + 2)^2 = x^2 + 4x + 4$$
$$(2a + 3b)^2 = 4a^2 + 12ab + 9b^2$$
Application for computations:$$52^2 = (50 + 2)^2 = 2500 + 200 + 4 = 2704$$$$101^2 = (100 + 1)^2 = 10000 + 200 + 1 = 10201$$
$$(a - b)^2 = a^2 - 2ab + b^2$$
In words: The square of the difference of two expressions equals the square of the first expression, minus twice the product of the first and the second, plus the square of the second expression.
Proof:$$(a - b)^2 = (a - b)(a - b) = a^2 - ab - ab + b^2 = a^2 - 2ab + b^2$$
Examples:$$(7 - 3)^2 = 49 - 42 + 9 = 16$$
check: $4^2 = 16$ ✓
$$(x - 5)^2 = x^2 - 10x + 25$$
$$(3a - 2b)^2 = 9a^2 - 12ab + 4b^2$$
Application for computations:$$49^2 = (50 - 1)^2 = 2500 - 100 + 1 = 2401$$$$98^2 = (100 - 2)^2 = 10000 - 400 + 4 = 9604$$
Note that:$$(a - b)^2 = (a + (-b))^2 = a^2 + 2a(-b) + (-b)^2 = a^2 - 2ab + b^2$$
$$a^2 - b^2 = (a - b)(a + b)$$
In words: The difference of the squares of two expressions equals the product of their difference and their sum.
Proof:$$(a - b)(a + b) = a \cdot a + a \cdot b - b \cdot a - b \cdot b = a^2 - b^2$$
Examples:$$9 - 4 = (3 - 2)(3 + 2) = 1 \cdot 5 = 5$$
$$x^2 - 1 = (x - 1)(x + 1)$$
$$4x^2 - 9y^2 = (2x)^2 - (3y)^2 = (2x - 3y)(2x + 3y)$$
Application for computations:$$99 \cdot 101 = (100 - 1)(100 + 1) = 100^2 - 1 = 10000 - 1 = 9999$$
$$48 \cdot 52 = (50 - 2)(50 + 2) = 2500 - 4 = 2496$$
$$a^{2n} - b^{2n} = (a^n - b^n)(a^n + b^n)$$
Examples:$$x^4 - 16 = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)$$
$$a^6 - b^6 = (a^3)^2 - (b^3)^2 = (a^3 - b^3)(a^3 + b^3)$$
$$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$
Alternative form:$$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$$
Proof:$$(a + b)^3 = (a + b)(a + b)^2 = (a + b)(a^2 + 2ab + b^2)$$$$= a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3$$$$= a^3 + 3a^2b + 3ab^2 + b^3$$
Examples:$$(x + 1)^3 = x^3 + 3x^2 + 3x + 1$$
$$(2 + 3)^3 = 8 + 36 + 54 + 27 = 125$$
check: $5^3 = 125$ ✓
$$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$
Alternative form:$$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$$
Proof:$$(a - b)^3 = (a - b)(a - b)^2 = (a - b)(a^2 - 2ab + b^2)$$$$= a^3 - 2a^2b + ab^2 - a^2b + 2ab^2 - b^3$$$$= a^3 - 3a^2b + 3ab^2 - b^3$$
Examples:$$(x - 2)^3 = x^3 - 6x^2 + 12x - 8$$
$$(5 - 1)^3 = 125 - 75 + 15 - 1 = 64$$
check: $4^3 = 64$ ✓
$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$
Proof:$$(a + b)(a^2 - ab + b^2) = a^3 - a^2b + ab^2 + a^2b - ab^2 + b^3 = a^3 + b^3$$
Examples:$$8 + 27 = 2^3 + 3^3 = (2 + 3)(4 - 6 + 9) = 5 \cdot 7 = 35$$
check: $8 + 27 = 35$ ✓
$$x^3 + 1 = (x + 1)(x^2 - x + 1)$$
$$8a^3 + 27b^3 = (2a)^3 + (3b)^3 = (2a + 3b)(4a^2 - 6ab + 9b^2)$$
$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$
Proof:$$(a - b)(a^2 + ab + b^2) = a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3 = a^3 - b^3$$
Examples:$$27 - 8 = 3^3 - 2^3 = (3 - 2)(9 + 6 + 4) = 1 \cdot 19 = 19$$
check: $27 - 8 = 19$ ✓
$$x^3 - 1 = (x - 1)(x^2 + x + 1)$$
$$27a^3 - 64b^3 = (3a)^3 - (4b)^3 = (3a - 4b)(9a^2 + 12ab + 16b^2)$$
Method: Find the GCD of the coefficients and the common variables, and take them outside the brackets.
Examples:$$6x^2 + 9x = 3x(2x + 3)$$
check: $3x \cdot 2x + 3x \cdot 3 = 6x^2 + 9x$ ✓
$$12a^2b - 8ab^2 + 4ab = 4ab(3a - 2b + 1)$$
$$15x^3y^2 + 10x^2y^3 - 5xy = 5xy(3x^2y + 2xy^2 - 1)$$
Method: Split the expression into groups, extract the common factor from each group, and then extract the common factor for all groups.
Examples:
Example 1:$$xy + 2x + 3y + 6$$$$= x(y + 2) + 3(y + 2)$$$$= (y + 2)(x + 3)$$
Example 2:$$a^3 + a^2b + ab^2 + b^3$$$$= a^2(a + b) + b^2(a + b)$$$$= (a + b)(a^2 + b^2)$$
Example 3:$$2ac + 2bc - a - b$$$$= 2c(a + b) - (a + b)$$$$= (a + b)(2c - 1)$$
Example 1:$$x^2 - 9 = x^2 - 3^2 = (x - 3)(x + 3)$$
Example 2:$$4a^2 + 4ab + b^2 = (2a)^2 + 2 \cdot 2a \cdot b + b^2 = (2a + b)^2$$
Example 3:$$8x^3 - 27 = (2x)^3 - 3^3 = (2x - 3)(4x^2 + 6x + 9)$$
Method 1: Via the discriminant
If $ax^2 + bx + c = 0$ has roots $x_1$ and $x_2$, then:$$ax^2 + bx + c = a(x - x_1)(x - x_2)$$
Example:$$x^2 - 5x + 6 = 0$$
Roots: $x = 2, x = 3$ (since $2 + 3 = 5$ and $2 \cdot 3 = 6$)
$$x^2 - 5x + 6 = (x - 2)(x - 3)$$
Method 2: "Splitting the middle term" (for $a \neq 1$)
For $ax^2 + bx + c$:1. Find two numbers $p$ and $q$ such that $pq = ac$ and $p + q = b$2. Split $bx = px + qx$3. Factor by grouping
Example:$$2x^2 + 7x + 3$$
We need $p$ and $q$ such that $pq = 2 \cdot 3 = 6$ and $p + q = 7$
These are $p = 6, q = 1$ (since $6 \cdot 1 = 6$ and $6 + 1 = 7$)
$$2x^2 + 7x + 3 = 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1)$$
General strategy:1. Extract the common factor2. Apply abbreviated multiplication formulas3. Cancel common factors4. Combine like terms
Example 1: Simplify $(x + 1)^2 - (x - 1)^2$
Method 1 — expansion via formulas:$$= (x^2 + 2x + 1) - (x^2 - 2x + 1)$$$$= x^2 + 2x + 1 - x^2 + 2x - 1$$$$= 4x$$
Method 2 — application of the difference of squares:$$= [(x+1) - (x-1)][(x+1) + (x-1)]$$$$= [2][2x]$$$$= 4x$$
Example 2: Simplify $\frac{a^3 - b^3}{a - b}$
$$= \frac{(a - b)(a^2 + ab + b^2)}{a - b}$$$$= a^2 + ab + b^2$$
(for $a \neq b$)
Method 1: Transform the left side into the right side
Example: Prove that $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$
$$(a + b + c)^2 = ((a + b) + c)^2$$$$= (a + b)^2 + 2(a + b)c + c^2$$$$= a^2 + 2ab + b^2 + 2ac + 2bc + c^2$$$$= a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$$ ✓
Method 2: Transform both sides to the same form
Example: Prove that $a^4 - b^4 = (a - b)(a + b)(a^2 + b^2)$
Left side:$$a^4 - b^4 = (a^2)^2 - (b^2)^2 = (a^2 - b^2)(a^2 + b^2)$$$$= (a - b)(a + b)(a^2 + b^2)$$
The right side is already in this form.
They are equal! ✓
Example 1: Compute $98 \cdot 102$
$$98 \cdot 102 = (100 - 2)(100 + 2) = 100^2 - 4 = 10000 - 4 = 9996$$
Example 2: Compute $31^2$
$$31^2 = (30 + 1)^2 = 900 + 60 + 1 = 961$$
Example 3: Compute $17^3$
$$17^3 = (20 - 3)^3 = 20^3 - 3 \cdot 20^2 \cdot 3 + 3 \cdot 20 \cdot 9 - 27$$$$= 8000 - 3600 + 540 - 27 = 4913$$
Example 1: Simplify $\frac{x^2 - 1}{x - 1}$
$$\frac{x^2 - 1}{x - 1} = \frac{(x-1)(x+1)}{x-1} = x + 1$$ (for $x \neq 1$)
Example 2: Simplify $\frac{a^3 + 8}{a + 2}$
$$\frac{a^3 + 8}{a + 2} = \frac{a^3 + 2^3}{a + 2} = \frac{(a+2)(a^2-2a+4)}{a+2} = a^2 - 2a + 4$$ (for $a \neq -2$)
Statement: Simplify $(a + b)^2 - (a - b)^2$
Solution:
Method 1:$$(a + b)^2 - (a - b)^2 = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2)$$$$= a^2 + 2ab + b^2 - a^2 + 2ab - b^2$$$$= 4ab$$
Method 2 (via difference of squares):$$(a+b)^2 - (a-b)^2 = [(a+b) - (a-b)][(a+b) + (a-b)]$$$$= (2b)(2a) = 4ab$$
Answer: $4ab$
Statement: Factor $x^3 - 8$
$$x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)$$
Answer: $(x - 2)(x^2 + 2x + 4)$
Statement: Factor $3x^2 + 8x + 4$
We need $p, q$ such that $pq = 3 \cdot 4 = 12$ and $p + q = 8$
These are $p = 6, q = 2$
$$3x^2 + 8x + 4 = 3x^2 + 6x + 2x + 4$$$$= 3x(x + 2) + 2(x + 2)$$$$= (x + 2)(3x + 2)$$
Answer: $(x + 2)(3x + 2)$
Statement: Compute $49^2 - 51^2$
$$49^2 - 51^2 = (49 - 51)(49 + 51) = (-2)(100) = -200$$
Answer: -200
Statement: Prove that $(a - b)^3 + (b - c)^3 + (c - a)^3 = 3(a-b)(b-c)(c-a)$
Note that $(a - b) + (b - c) + (c - a) = 0$
We use the property: If $x + y + z = 0$, then $x^3 + y^3 + z^3 = 3xyz$
Let $x = a - b, y = b - c, z = c - a$
Then: $x + y + z = 0$
Therefore:$$(a-b)^3 + (b-c)^3 + (c-a)^3 = 3(a-b)(b-c)(c-a)$$ ✓
Wrong: $(a + b)^2 = a^2 + b^2$
Correct: $(a + b)^2 = a^2 + 2ab + b^2$
Don't forget the middle term $2ab$!
Wrong: $a^2 - b^2 = (a - b)^2$
Correct: $a^2 - b^2 = (a - b)(a + b)$
Wrong: $a^3 + b^3 = (a + b)^3$
Correct: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Wrong: $x^2 + 5x + 6 = (x + 2)(x + 4)$ (incorrect numbers)
Correct: $x^2 + 5x + 6 = (x + 2)(x + 3)$ (since $2 + 3 = 5$ and $2 \cdot 3 = 6$)
Wrong: $x^2 + 2x = x(x + 2)$ then on expansion we get $x^2 + 2x$, all correct, but...
If it was required to extract $2x$:Wrong: $2x^2 + 4x = x(2x + 4)$ (only $x$ was extracted, which is incorrect)
Correct: $2x^2 + 4x = 2x(x + 2)$
Wrong: $\frac{x^2 + 2x}{x} = x + 2x = 3x$ (the terms are counted incorrectly!)
Correct: $\frac{x^2 + 2x}{x} = \frac{x(x+2)}{x} = x + 2$ (for $x \neq 0$)
The square of a sum $(a+b)^2$ is the area of a square with side $(a+b)$:- Two squares of area $a^2$ and $b^2$- Two rectangles of area $ab$ each
On the exam this saves time!
This often simplifies further work.
After factoring, always expand the brackets to make sure the result is correct.
When cancelling fractions, indicate for which values of the variable this is valid.
For example: $\frac{x^2-1}{x-1} = x+1$ for $x \neq 1$
All course topics · Practice problems
Please enable JavaScript to use the practice app.