Comprehensive Guide to Factoring Polynomials

The fundamental step in polynomial simplification is identifying the greatest common factor (GCF). This process involves extracting the largest numerical coefficient and the highest power of each variable that divides evenly into every term of the expression.
Example 1: $2x^2+18x$
- First, assess the numerical coefficients $2$ and $18$ . The greatest common divisor is $2$ .
- Next, assess the variables $x^2$ and $x$ . The greatest common factor is $x$ .
- Initial extraction: $2(x^2 + 9x)$ .
- Complete extraction of GCF: $2x(x + 9)$ .
- Note: In some procedural steps, terms might be grouped as $2(x+3)(x+3)$ if misidentified as a perfect square, but the standard GCF result is $2x(x + 9)$ .

Polynomials in the form of $a^2 - b^2$ are factored using the formula $(a - b)(a + b)$ . This applies when two perfect squares are being subtracted.
Example 2: $t^2 - 144$
- Identify the square roots: $\sqrt{t^2} = t$ and $\sqrt{144} = 12$ .
- Apply the identity: $(t - 12)(t + 12)$ .
Example 3: $81a^2 - 49$
- Identify the square roots: $\sqrt{81a^2} = 9a$ and $\sqrt{49} = 7$ .
- Apply the identity: $(9a - 7)(9a + 7)$ .

When the leading coefficient is $1$ , find two integers that multiply to give the constant term $c$ and add together to give the middle coefficient $b$ .
Example 4: $y^2 - y - 20$
- Required: Factors of $-20$ that sum to $-1$ .
- The integers are $-5$ and $4$ .
- Factored form: $(y - 5)(y + 4)$ .
Example 5: $h^2 + 13h + 30$
- Required: Factors of $30$ that sum to $13$ .
- The integers are $10$ and $3$ .
- Factored form: $(h + 10)(h + 3)$ .

When the leading coefficient $a$ is not $1$ , specific methods such as the "ac" method (decomposition) are used. Multiply $a$ and $c$ , then find factors of that product that sum to $b$ .
Example 6: $2e^2 + 11e + 15$
- Calculate $ac = 2 \times 15 = 30$ .
- Find factors of $30$ that sum to $11$ : $5$ and $6$ .
- Decomposition: $2e^2 + 6e + 5e + 15$ .
- Grouping: $2e(e + 3) + 5(e + 3)$ .
- Factored form: $(2e + 5)(e + 3)$ .
Example 7: $12z^2 - z - 1$
- Calculate $ac = 12 \times (-1) = -12$ .
- Find factors of $-12$ that sum to $-1$ : $-4$ and $3$ .
- Decomposition: $12z^2 - 4z + 3z - 1$ .
- Grouping: $4z(3z - 1) + 1(3z - 1)$ .
- Factored form: $(4z + 1)(3z - 1)$ .
Example 8: $10u^2 - 3u - 18$
- Calculate $ac = 10 \times (-18) = -180$ .
- Find factors of $-180$ that sum to $-3$ : $-15$ and $12$ .
- Decomposition: $10u^2 - 15u + 12u - 18$ .
- Grouping: $5u(2u - 3) + 6(2u - 3)$ .
- Factored form: $(5u + 6)(2u - 3)$ .

A perfect square trinomial follows the pattern $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$ .
Example 9: $r^2 + 10r + 25$
- Observe the first term $\sqrt{r^2} = r$ and last term $\sqrt{25} = 5$ .
- Verify the middle term: $2 \times r \times 5 = 10r$ .
- Factored form: $(r + 5)^2 \text{ or } (r + 5)(r + 5)$ .
Example 10: $16t^2 - 40t + 25$
- Observe the first term $\sqrt{16t^2} = 4t$ and last term $\sqrt{25} = 5$ .
- Verify the middle term: $2 \times 4t \times 5 = 40t$ . Since the middle term is negative, use the subtraction identity.
- Factored form: $(4t - 5)^2$ .

Complete factoring requires checking for a GCF first, then applying specific formulas (difference of squares, trinomial factoring) to the remaining expression.
Example 11: $2a^2 - 50$
- Step 1 (GCF): Factor out $2$ to get $2(a^2 - 25)$ .
- Step 2 (Difference of Squares): Factor $(a^2 - 25)$ to get $(a - 5)(a + 5)$ .
- Complete result: $2(a - 5)(a + 5)$ .
Example 12: $4x^3 + 23x^2 + 15x$
- Step 1 (GCF): Factor out $x$ to get $x(4x^2 + 23x + 15)$ .
- Step 2 (Trinomial Factoring): Solve for $4x^2 + 23x + 15$ .
  - $ac = 60$ . Factors of $60$ that sum to $23$ are $20$ and $3$ .
  - Decompose: $4x^2 + 20x + 3x + 15$ .
  - Group: $4x(x + 5) + 3(x + 5)$ .
  - Resulting binomials: $(4x + 3)(x + 5)$ .
- Complete result: $x(4x + 3)(x + 5)$ .
Example 13: $3h^3 - 27h$
- Step 1 (GCF): Factor out $3h$ to get $3h(h^2 - 9)$ .
- Step 2 (Difference of Squares): Factor $(h^2 - 9)$ to get $(h - 3)(h + 3)$ .
- Complete result: $3h(h - 3)(h + 3)$ .