Methods of Factorization: Finding the GCF

Finding the Greatest Common Factor (GCF) is your first step in factoring expressions. It's like identifying what several terms share in common.

To find the GCF of numbers, break them down into their prime factors. For example, with 24 and 36, write 24 = 2×2×2×3 and 36 = 2×2×3×3. The common factors (2×2×3 = 12) give us the GCF.

For algebraic expressions like 7x³, 14x², and 21x⁴, find the GCF in two parts:

1. Find the GCF of the coefficients (7 in this case)
2. Find the variable with the lowest power (x² here) So the GCF is 7x².

Pro Tip: Always look for the GCF first before trying other factoring methods - it makes the next steps much easier!

When expressions contain binomials like 3x$a+b$ and 2y$a+b$, the common factor is simply $a+b$. Remember that your goal is to pull out everything that's common to all terms.

Factoring Out the GCF in Complex Expressions

Factoring gets more interesting with longer expressions. The key is to identify what's common to all terms and pull it out.

With expressions like 15y³+12y⁴, start by finding the GCF - here it's 3y³. Then rewrite as 3y³$5+4y$. For 9a⁴b-18a⁵b+27ab, the GCF is 9ab, giving us 9ab$1-2a+3a²$.

When you see expressions like 2w$x+3$-5$x+3$, look for the common binomial factor. Here $x+3$ appears in both terms, so we can factor it out: $x+3$$2w-5$.

More complex examples require careful grouping. For 8ax+12a+2bx+8b, you might not immediately see a common factor. Try regrouping: 4a$2x+3$+2b$x+4$, which reveals $x+4$ is common to both groups, giving the final factored form $3a+2b$$x+4$.

Remember: Sometimes you need to factor multiple times. In 16w⁴-40w³-12w²+30w, first factor out 2w to get 2w$8w³-20w²-6w+15$, then look for other patterns in what remains.

Factoring Trinomials and Special Products

The A.C. method is super helpful for factoring trinomials like ax²+bx+c:

1. Find the product of a and c (that's ac)
2. Look for two numbers that multiply to give ac and add up to b
3. Split the middle term using these numbers
4. Group and factor

For example, with 2x²+7x+6:

• ac = 12
• We need numbers that multiply to 12 and add to 7 (that's 3 and 4)
• Rewrite as 2x²+4x+3x+6
• Factor by grouping: $2x+3$$x+2$

Difference of squares follows the pattern a²-b²=$a+b$$a-b$. So y²-25 = $y+5$$y-5$. But remember: sum of squares $like p²+9$ cannot be factored with real numbers.

Watch out! Don't try to factor sum of squares like a²+b² - they're prime over real numbers and can't be factored further!

For complex expressions like w⁴-81, recognize that it's (w²)²-9² and factor it as $w²+9$$w²-9$, then factor w²-9 again to get $w²+9$$w+3$$w-3$.

Perfect Square Trinomials and Advanced Factoring

Perfect square trinomials follow these patterns:

• a²+2ab+b² = $a+b$²
• a²-2ab+b² = $a-b$²

To spot them, check if:

1. The first and last terms are perfect squares
2. The middle term equals 2ab or -2ab

For example, x²+14x+49 is a perfect square trinomial because:

• First term is x² (perfect square)
• Last term is 49 = 7² (perfect square)
• Middle term is 14x, which equals 2(x)(7)
• Therefore, x²+14x+49 = $x+7$²

Another example: 25y²-20y+4 = $5y-2$² because:

• 25y² = (5y)²
• 4 = 2²
• -20y = -2(5y)(2)

When factoring complex expressions like 18c³-48c²d+32cd², look for a common factor first. Here we have 2c, so we get 2c$9c²-24cd+16d²$, which reveals a perfect square trinomial: 2c$3c-4d$².

Smart approach: Always check if your trinomial might be a perfect square - it makes factoring much faster than using other methods!

Sum and Difference of Cubes

The sum of cubes follows the pattern: a³+b³ = $a+b$$a²-ab+b²$

The difference of cubes follows: a³-b³ = $a-b$$a²+ab+b²$

These patterns are super useful for expressions like w³+64:

• Recognize that 64 = 4³
• Apply the formula: w³+4³ = $w+4$$w²-4w+16$

For more complex examples like 27p³-1000q³:

• Rewrite as (3p)³-(10q)³
• Apply the difference of cubes formula: $3p-10q$$9p²+30pq+100q²$

These factoring techniques work together. For 3y⁴-48:

1. Factor out common factor: 3$y⁴-16$
2. Recognize y⁴-16 as a difference of squares: $y²-4$$y²+4$
3. Factor y²-4 as another difference of squares: $y-2$$y+2$
4. Final answer: 3$y-2$$y+2$$y²+4$

Factoring strategy: When you see expressions with cubes or higher powers, think about using these special patterns - they save a ton of work compared to other methods!