Step 1: Start with the Smallest Prime

Begin by testing whether your number is divisible by 2, the smallest prime number. If it divides evenly, write down 2 as a factor and divide your number by 2.

Example: For 180, we see that 180 ÷ 2 = 90, so 2 is our first prime factor.

Step 2: Continue with the Result

Take the quotient from Step 1 and repeat the process. Keep dividing by 2 as long as the result is even. When you can no longer divide by 2, move to the next prime number (3).

Example: 90 ÷ 2 = 45. Now 45 is odd, so we move to 3. Since 45 ÷ 3 = 15, we have another factor of 3.

Step 3: Test Each Prime in Order

Continue testing prime numbers (2, 3, 5, 7, 11, etc.) in order. For each prime that divides evenly, record it as a factor and divide the result.

Example: 15 ÷ 3 = 5, and 5 is prime, so we stop.

Step 4: Stop When You Reach a Prime

The process ends when your quotient is itself a prime number. This final number is the last factor.

Example: Our factors are 2, 2, 3, 3, and 5. Therefore, 180 = 2² × 3² × 5

Step 1: Write the Number at the Top

Start by writing your number at the top of your paper. This is the "trunk" of your factor tree.

Step 2: Split into Two Factors

Find any two numbers that multiply to give your starting number. Draw branches down to these two factors.

Example: For 180, you might choose 18 × 10, or 20 × 9, or any other factor pair.

Step 3: Continue Breaking Down Composite Numbers

For each composite (non-prime) number in your tree, split it into two more factors. Circle any prime numbers you encounter.

Step 4: Collect the Primes

When all branches end in prime numbers, collect all the circled primes. These are your prime factors.

Example: No matter which branches you chose, 180 will always end up as 2 × 2 × 3 × 3 × 5.

Step 1: Find Prime Factorization of Each Number

Start by finding the complete prime factorization of each number, expressing them with exponents.

Example: To find LCM(12, 18):

• 12 = 2² × 3
• 18 = 2 × 3²

Step 2: Identify All Prime Factors

List every prime number that appears in any of the factorizations. Don't worry about how many times each prime appears yet.

Example: The primes that appear are 2 and 3.

Step 3: Take the Highest Power of Each Prime

For each prime on your list, look at all the factorizations and select the highest exponent (power) that appears for that prime.

Example:

• For prime 2: appears as 2² in 12 and 2¹ in 18, so take 2²
• For prime 3: appears as 3¹ in 12 and 3² in 18, so take 3²

Step 4: Multiply the Results

Multiply together all the highest powers you identified. The result is your LCM.

Example: LCM = 2² × 3² = 4 × 9 = 36

The Process is the Same

To find the LCM of three or more numbers, follow the exact same steps: find all prime factorizations, identify all primes, and take the highest power of each.

Example: Find LCM(8, 12, 15):

• 8 = 2³
• 12 = 2² × 3
• 15 = 3 × 5

All primes: 2, 3, 5
Highest powers: 2³, 3¹, 5¹
LCM = 2³ × 3 × 5 = 8 × 3 × 5 = 120

Step 1: Find Prime Factorization of Each Number

Just like with LCM, start by finding the complete prime factorization of each number.

Example: To find GCF(48, 60):

• 48 = 2⁴ × 3
• 60 = 2² × 3 × 5

Step 2: Identify Common Prime Factors

List only the prime numbers that appear in ALL of the factorizations. If a prime appears in one number but not another, it's not common.

Example: Both 48 and 60 contain 2 and 3, but only 60 has 5. So the common primes are 2 and 3.

Step 3: Take the Lowest Power of Each Common Prime

For each common prime, look at its exponent in each factorization and select the smallest exponent.

Example:

• For prime 2: appears as 2⁴ in 48 and 2² in 60, so take 2²
• For prime 3: appears as 3¹ in both, so take 3¹

Step 4: Multiply the Results

Multiply together the lowest powers of the common primes. The result is your GCF.

Example: GCF = 2² × 3 = 4 × 3 = 12

Using GCF to Simplify

To simplify a fraction, divide both the numerator and denominator by their GCF.

Example: Simplify 48/60:

• GCF(48, 60) = 12
• 48 ÷ 12 = 4
• 60 ÷ 12 = 5
• Therefore, 48/60 = 4/5

Step 1: Set Up the Pattern

You're looking for two numbers that multiply to give c and add to give b. The factored form will be (x + m)(x + n) where m and n are these two numbers.

Example: Factor x² + 7x + 12

Need two numbers that multiply to 12 and add to 7.

Step 2: List Factor Pairs of c

Write down all pairs of numbers that multiply to give c.

Example: Factors of 12: (1,12), (2,6), (3,4)

Step 3: Find the Pair That Sums to b

Check which pair adds up to b. Pay attention to signs.

Example: 3 + 4 = 7 ✓

So x² + 7x + 12 = (x + 3)(x + 4)

Step 4: Solve Each Factor

Set each factor equal to zero and solve.

Example:

• x + 3 = 0 → x = -3
• x + 4 = 0 → x = -4

Step 1: Multiply a × c

Calculate the product of the first and last coefficients.

Example: Factor 6x² + 11x + 4

a × c = 6 × 4 = 24

Step 2: Find Factors That Sum to b

Find two numbers that multiply to (a × c) and add to b.

Example: Need factors of 24 that sum to 11. The pair is 3 and 8.

Step 3: Rewrite the Middle Term

Split the middle term (bx) into two terms using your found numbers.

Example: 6x² + 11x + 4 becomes 6x² + 3x + 8x + 4

Step 4: Factor by Grouping

Group the first two terms and the last two terms, then factor out the GCF from each group.

Example:

• (6x² + 3x) + (8x + 4)
• 3x(2x + 1) + 4(2x + 1)
• (3x + 4)(2x + 1)

Difference of Squares: a² - b²

Pattern: a² - b² = (a + b)(a - b)

Example: x² - 25 = (x + 5)(x - 5)

This pattern only works when you have subtraction and both terms are perfect squares.

Perfect Square Trinomials

Pattern: a² + 2ab + b² = (a + b)²
Pattern: a² - 2ab + b² = (a - b)²

Example: x² + 6x + 9 = (x + 3)²

Check: The first and last terms are perfect squares, and the middle term is twice their product.

Step 1: Find the Prime Factorization

Write the radicand as a product of prime factors, using exponents.

Example: Simplify √180

180 = 2² × 3² × 5

Step 2: Group Perfect Squares

For square roots, look for prime factors that appear in pairs (exponent of 2 or more).

Example: √(2² × 3² × 5) - we have two perfect squares: 2² and 3²

Step 3: Extract Perfect Squares

For each pair of identical factors, bring one factor outside the radical.

Example: √(2² × 3² × 5) = 2 × 3 × √5 = 6√5

Step 4: Verify Simplification

Your radical is simplified when the radicand has no perfect square factors left.

Example: 6√5 is fully simplified because 5 is prime and has no perfect square factors.

Group by Index

For cube roots (index 3), look for prime factors that appear three times. For fourth roots (index 4), look for groups of four.

Example: Simplify ∛(216)

• 216 = 2³ × 3³
• Both primes appear in groups of 3
• ∛(2³ × 3³) = 2 × 3 = 6

Mixed Exponents

When exponents don't divide evenly by the index, extract what you can and leave the rest inside.

Example: Simplify ∛(108)

• 108 = 2² × 3³
• 3³ can come out (as 3), but 2² stays inside
• ∛(2² × 3³) = 3∛4

Step 1: Identify the Denominator Radical

Look at what radical appears in the denominator.

Example: Simplify √3/√8

Step 2: Simplify First

Before rationalizing, simplify any radicals you can.

Example: √8 = √(2² × 2) = 2√2

So we have √3/(2√2)

Step 3: Multiply by the Conjugate

Multiply both numerator and denominator by whatever radical is in the denominator.

Example: (√3/(2√2)) × (√2/√2) = √6/(2 × 2) = √6/4

Separate Numerator and Denominator

Use the property √(a/b) = √a/√b to split the radical.

Example: √(3/8) = √3/√8

Then simplify and rationalize as shown above to get √6/4.

Step 1: Start with Standard Form

Begin with your equation in the form y = x² + bx + c.

Example: Convert y = x² + 6x + 5 to vertex form

Step 2: Calculate (b/2)²

Take the coefficient of x, divide it by 2, then square the result. This is the number needed to complete the square.

Example: b = 6, so (6/2)² = 3² = 9

Step 3: Add and Subtract (b/2)²

Add this number after the x term, then immediately subtract it to keep the equation balanced.

Example: y = x² + 6x + 9 - 9 + 5

Step 4: Factor the Perfect Square

The first three terms (x² + bx + (b/2)²) form a perfect square trinomial. Factor it.

Example: y = (x + 3)² - 9 + 5 = (x + 3)² - 4

Step 5: Identify the Vertex

Compare to y = a(x - h)² + k. Remember the minus sign in the formula.

Example: y = (x - (-3))² + (-4), so vertex is (-3, -4)

Step 1: Factor Out a

Factor the coefficient a from the x² and x terms only (not the constant).

Example: Convert y = 2x² - 8x + 3

y = 2(x² - 4x) + 3

Step 2: Complete the Square Inside the Parentheses

Calculate (b/2)² using the new b value inside the parentheses.

Example: Inside the parentheses, b = -4, so (-4/2)² = 4

y = 2(x² - 4x + 4 - 4) + 3

Step 3: Distribute the Subtracted Term

The subtracted (b/2)² is inside the parentheses, so when you pull it out, multiply by a.

Example: y = 2(x² - 4x + 4) + 2(-4) + 3

y = 2(x² - 4x + 4) - 8 + 3

Step 4: Factor and Simplify

Factor the perfect square trinomial and combine constants.

Example: y = 2(x - 2)² - 5

Vertex: (2, -5)

Step 1: Start with the General Form

Begin with the standard quadratic equation where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).

Equation: \[ax^2 + bx + c = 0\]

Step 2: Isolate the Variable Terms

Move the constant term to the right side of the equation by subtracting \(c\) from both sides.

Result: \[ax^2 + bx = -c\]

This prepares us to complete the square on the left side.

Step 3: Factor Out the Leading Coefficient

To complete the square when \(a \neq 1\), we must factor out the coefficient \(a\) from both \(x^2\) and \(x\) terms.

Result: \[a\left(x^2 + \frac{b}{a}x\right) = -c\]

This makes the coefficient of \(x^2\) equal to 1 inside the parentheses, which is necessary for completing the square.

Step 4: Complete the Square

To complete the square, take half of the coefficient of \(x\) (which is \(\frac{b}{a}\)), square it, and add it inside the parentheses. Remember: whatever we add inside must be balanced.

Half of \(\frac{b}{a}\) is \(\frac{b}{2a}\), and squaring gives \(\left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}\)

Add this inside the parentheses, but since it's multiplied by \(a\), we're actually adding \(a \cdot \frac{b^2}{4a^2} = \frac{b^2}{4a}\) to the left side.

Result: \[a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}\right) = -c + \frac{b^2}{4a}\]

Step 5: Factor the Perfect Square Trinomial

The expression inside the parentheses is now a perfect square trinomial. It factors as \(\left(x + \frac{b}{2a}\right)^2\).

Result: \[a\left(x + \frac{b}{2a}\right)^2 = -c + \frac{b^2}{4a}\]

This is the key step: we've transformed the left side into a perfect square.

Step 6: Combine the Right Side

Combine the terms on the right side by finding a common denominator. The common denominator is \(4a\).

\(-c = \frac{-4ac}{4a}\), so: \[-c + \frac{b^2}{4a} = \frac{-4ac + b^2}{4a} = \frac{b^2 - 4ac}{4a}\]

Result: \[a\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a}\]

Step 7: Divide Both Sides by a

Divide both sides by \(a\) to isolate the squared term.

Result: \[\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}\]

Step 8: Take the Square Root of Both Sides

Take the square root of both sides. Remember that taking a square root introduces both positive and negative solutions (\(\pm\)).

Result: \[x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}}\]

Simplify the square root: \(\sqrt{\frac{b^2 - 4ac}{4a^2}} = \frac{\sqrt{b^2 - 4ac}}{2a}\)

Simplified: \[x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}\]

Step 9: Solve for x

Subtract \(\frac{b}{2a}\) from both sides to isolate \(x\).

Result: \[x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}\]

Step 10: Combine into Final Formula

Combine the terms over the common denominator \(2a\).

The Quadratic Formula:

This formula gives both solutions to any quadratic equation \(ax^2 + bx + c = 0\) (where \(a \neq 0\)).

Solving \(x^2 + 6x + 5 = 0\) by Completing the Square

Let's apply the same steps to a specific equation to see the derivation in action.

• Start: \(x^2 + 6x + 5 = 0\)
• Isolate variables: \(x^2 + 6x = -5\)
• Complete the square: Half of 6 is 3, and \(3^2 = 9\)
  Add 9: \(x^2 + 6x + 9 = -5 + 9\)
• Factor: \((x + 3)^2 = 4\)
• Take square root: \(x + 3 = \pm 2\)
• Solve: \(x = -3 \pm 2\)
• Solutions: \(x = -1\) or \(x = -5\)

You can verify this with the quadratic formula: \[x = \frac{-6 \pm \sqrt{36 - 20}}{2} = \frac{-6 \pm 4}{2} = -1 \text{ or } -5 \quad \checkmark\]

Step 1: Arrange in Standard Form

Write both polynomials in descending order of degree (highest power first). If any degrees are missing, include them with a coefficient of 0.

Example: Divide (x³ + 5x² + 8x + 4) by (x + 2)

Both are already in standard form.

Step 2: Set Up the Division

Write it like long division: divisor on the left, dividend under the division symbol.

Step 1: Divide the Leading Terms

Divide the first term of the dividend by the first term of the divisor. This gives you the first term of the quotient.

Example: x³ ÷ x = x²

Write x² above the division symbol.

Step 2: Multiply and Subtract

Multiply the entire divisor by the quotient term you just found. Write the result below the dividend and subtract.

Example: x² × (x + 2) = x³ + 2x²

Subtract from x³ + 5x²: (x³ + 5x²) - (x³ + 2x²) = 3x²

Step 3: Bring Down the Next Term

Bring down the next term from the dividend to continue the process.

Example: Bring down the 8x to get 3x² + 8x

Step 4: Repeat the Process

Repeat steps 1-3 with the new polynomial.

Example:

• 3x² ÷ x = 3x (second term of quotient)
• 3x × (x + 2) = 3x² + 6x
• Subtract: (3x² + 8x) - (3x² + 6x) = 2x
• Bring down 4: 2x + 4

Step 5: Continue Until Complete

Keep repeating until the degree of the remainder is less than the degree of the divisor.

Example:

• 2x ÷ x = 2 (third term of quotient)
• 2 × (x + 2) = 2x + 4
• Subtract: (2x + 4) - (2x + 4) = 0

Remainder is 0, so we're done.

Expressing the Answer

When there's a remainder, express the answer as: Quotient + Remainder/Divisor

Example: If dividing (2x³ - 3x² + 4x - 5) by (x - 1) gives quotient 2x² - x + 3 and remainder -2:

Answer: 2x² - x + 3 - 2/(x - 1)

Fill in Zeros

If the dividend is missing a degree, include it with coefficient 0.

Example: To divide x³ + 4 by x + 1, rewrite as:

(x³ + 0x² + 0x + 4) ÷ (x + 1)

This keeps your columns aligned properly.

Step 1: Identify the Key Values

Identify the first term (a₁), the last term (aₙ), and the common difference (d).

Example: Find the sum of 3 + 6 + 9 + 12 + ... + 99

• First term (a₁) = 3
• Last term (aₙ) = 99
• Common difference (d) = 3

Step 2: Find the Number of Terms

Use the formula: \(n = \frac{\text{last term} - \text{first term}}{\text{common difference}} + 1\)

Example: \(n = \frac{99 - 3}{3} + 1 = \frac{96}{3} + 1 = 32 + 1 = 33\) terms

Why add 1? Because we're counting inclusive endpoints. From 3 to 99 by threes gives us 33 terms, not 32.

Step 3: Calculate the Average

The average of an arithmetic sequence is simply the mean of the first and last terms.

Formula: \(\text{Average} = \frac{\text{first term} + \text{last term}}{2}\)

Example: \(\text{Average} = \frac{3 + 99}{2} = \frac{102}{2} = 51\)

Step 4: Calculate the Sum

Multiply the average by the number of terms.

Formula: \(\text{Sum} = \text{Average} \times n\)

Example: \(\text{Sum} = 51 \times 33 = 1{,}683\)

Finding the Last Term

If you know the first term, common difference, and number of terms, you can find the last term:

Formula: \(\text{last term} = \text{first term} + (n - 1) \times d\)

Example: What is the 50th term in the sequence 5, 8, 11, 14, ...?

\(a_{50} = 5 + (50 - 1) \times 3 = 5 + 147 = 152\)

Sum of First n Natural Numbers

The series 1 + 2 + 3 + ... + n is the most famous arithmetic series.

Formula: \(\text{Sum} = \frac{n(n + 1)}{2}\)

This is exactly what Gauss discovered! For n = 100: \(\frac{100 \times 101}{2} = 5{,}050\)

Sum of First n Odd Numbers

The series 1 + 3 + 5 + ... + (2n - 1) has a beautiful result:

Formula: \(\text{Sum} = n^2\)

For example, 1 + 3 + 5 + 7 + 9 = 25 = 5². The first 5 odd numbers sum to 5²!

Sum of First n Even Numbers

The series 2 + 4 + 6 + ... + 2n:

Formula: \(\text{Sum} = n(n + 1)\)

For example, 2 + 4 + 6 + 8 + 10 = 30 = 5 × 6.