Factor Polynomials - Ninth Grade Math
Introduction to Factoring
Factoring: The process of writing a polynomial as a product of its factors
Factor: A number or expression that divides evenly into another
Prime Factorization: Writing a number as a product of prime numbers
Why Factor?
• Simplify expressions
• Solve equations
• Find zeros of functions
• Identify patterns in algebra
Factor: A number or expression that divides evenly into another
Prime Factorization: Writing a number as a product of prime numbers
Why Factor?
• Simplify expressions
• Solve equations
• Find zeros of functions
• Identify patterns in algebra
Factoring is the REVERSE of multiplying:
• Multiply: $(x + 3)(x + 5) = x^2 + 8x + 15$
• Factor: $x^2 + 8x + 15 = (x + 3)(x + 5)$
• Multiply: $(x + 3)(x + 5) = x^2 + 8x + 15$
• Factor: $x^2 + 8x + 15 = (x + 3)(x + 5)$
1. GCF of Monomials
Greatest Common Factor (GCF): The largest factor that divides evenly into all given monomials
Also called: Greatest Common Divisor (GCD) or Highest Common Factor (HCF)
Also called: Greatest Common Divisor (GCD) or Highest Common Factor (HCF)
Finding GCF of Monomials:
Step 1: Find GCF of coefficients (numbers)
Step 2: For each variable, take the LOWEST power
Step 3: Multiply results from Steps 1 and 2
Rule for Variables:
$$\text{GCF}(x^m, x^n) = x^{\min(m,n)}$$
Take the smaller exponent
Step 1: Find GCF of coefficients (numbers)
Step 2: For each variable, take the LOWEST power
Step 3: Multiply results from Steps 1 and 2
Rule for Variables:
$$\text{GCF}(x^m, x^n) = x^{\min(m,n)}$$
Take the smaller exponent
Example 1: Find GCF of $12x^3$ and $18x^2$
Step 1: GCF of 12 and 18
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
GCF = 6
Step 2: GCF of $x^3$ and $x^2$
Take lower power: $x^2$
Step 3: Multiply: $6 \times x^2 = 6x^2$
Answer: GCF = $6x^2$
Step 1: GCF of 12 and 18
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
GCF = 6
Step 2: GCF of $x^3$ and $x^2$
Take lower power: $x^2$
Step 3: Multiply: $6 \times x^2 = 6x^2$
Answer: GCF = $6x^2$
Example 2: Find GCF of $15x^2y^5$, $12x^4y$, and $9x^3y^2$
Coefficients: GCF(15, 12, 9) = 3
x terms: GCF($x^2$, $x^4$, $x^3$) = $x^2$ (lowest power)
y terms: GCF($y^5$, $y$, $y^2$) = $y$ (lowest power)
Answer: GCF = $3x^2y$
Coefficients: GCF(15, 12, 9) = 3
x terms: GCF($x^2$, $x^4$, $x^3$) = $x^2$ (lowest power)
y terms: GCF($y^5$, $y$, $y^2$) = $y$ (lowest power)
Answer: GCF = $3x^2y$
Example 3: Find GCF of $24a^5b^3$ and $36a^2b^6$
GCF(24, 36) = 12
GCF($a^5$, $a^2$) = $a^2$
GCF($b^3$, $b^6$) = $b^3$
Answer: $12a^2b^3$
GCF(24, 36) = 12
GCF($a^5$, $a^2$) = $a^2$
GCF($b^3$, $b^6$) = $b^3$
Answer: $12a^2b^3$
2. LCM of Monomials
Least Common Multiple (LCM): The smallest monomial that is divisible by all given monomials
Finding LCM of Monomials:
Step 1: Find LCM of coefficients (numbers)
Step 2: For each variable, take the HIGHEST power
Step 3: Multiply results from Steps 1 and 2
Rule for Variables:
$$\text{LCM}(x^m, x^n) = x^{\max(m,n)}$$
Take the larger exponent
Step 1: Find LCM of coefficients (numbers)
Step 2: For each variable, take the HIGHEST power
Step 3: Multiply results from Steps 1 and 2
Rule for Variables:
$$\text{LCM}(x^m, x^n) = x^{\max(m,n)}$$
Take the larger exponent
Example 1: Find LCM of $4x^2y$ and $6xy^3$
Step 1: LCM(4, 6) = 12
Step 2:
• LCM($x^2$, $x$) = $x^2$ (highest power)
• LCM($y$, $y^3$) = $y^3$ (highest power)
Step 3: $12 \times x^2 \times y^3 = 12x^2y^3$
Answer: LCM = $12x^2y^3$
Step 1: LCM(4, 6) = 12
Step 2:
• LCM($x^2$, $x$) = $x^2$ (highest power)
• LCM($y$, $y^3$) = $y^3$ (highest power)
Step 3: $12 \times x^2 \times y^3 = 12x^2y^3$
Answer: LCM = $12x^2y^3$
Example 2: Find LCM of $3x^3y^2$ and $4x^5y^4$
LCM(3, 4) = 12
LCM($x^3$, $x^5$) = $x^5$
LCM($y^2$, $y^4$) = $y^4$
Answer: $12x^5y^4$
LCM(3, 4) = 12
LCM($x^3$, $x^5$) = $x^5$
LCM($y^2$, $y^4$) = $y^4$
Answer: $12x^5y^4$
Example 3: Find LCM of $10a^2b$, $15ab^3$, and $6a^3b^2$
LCM(10, 15, 6) = 30
LCM($a^2$, $a$, $a^3$) = $a^3$
LCM($b$, $b^3$, $b^2$) = $b^3$
Answer: $30a^3b^3$
LCM(10, 15, 6) = 30
LCM($a^2$, $a$, $a^3$) = $a^3$
LCM($b$, $b^3$, $b^2$) = $b^3$
Answer: $30a^3b^3$
Quick Comparison:
• GCF: Take LOWEST power of each variable
• LCM: Take HIGHEST power of each variable
• GCF: Take LOWEST power of each variable
• LCM: Take HIGHEST power of each variable
3. Factor Out a Monomial
Factoring Out GCF: Dividing each term by the GCF and writing the polynomial as a product
General Form:
$$ab + ac = a(b + c)$$
Steps to Factor Out Monomial:
Step 1: Find the GCF of all terms
Step 2: Divide each term by the GCF
Step 3: Write as: GCF(result from Step 2)
Step 4: Check by multiplying back
$$ab + ac = a(b + c)$$
Steps to Factor Out Monomial:
Step 1: Find the GCF of all terms
Step 2: Divide each term by the GCF
Step 3: Write as: GCF(result from Step 2)
Step 4: Check by multiplying back
Example 1: Factor $6x^2 + 9x$
Step 1: GCF(6x², 9x) = 3x
Step 2: Divide each term by 3x:
• $\frac{6x^2}{3x} = 2x$
• $\frac{9x}{3x} = 3$
Step 3: Write: $3x(2x + 3)$
Check: $3x(2x + 3) = 6x^2 + 9x$ ✓
Answer: $3x(2x + 3)$
Step 1: GCF(6x², 9x) = 3x
Step 2: Divide each term by 3x:
• $\frac{6x^2}{3x} = 2x$
• $\frac{9x}{3x} = 3$
Step 3: Write: $3x(2x + 3)$
Check: $3x(2x + 3) = 6x^2 + 9x$ ✓
Answer: $3x(2x + 3)$
Example 2: Factor $12x^3 - 18x^2 + 6x$
GCF: 6x
Divide:
• $\frac{12x^3}{6x} = 2x^2$
• $\frac{-18x^2}{6x} = -3x$
• $\frac{6x}{6x} = 1$
Answer: $6x(2x^2 - 3x + 1)$
GCF: 6x
Divide:
• $\frac{12x^3}{6x} = 2x^2$
• $\frac{-18x^2}{6x} = -3x$
• $\frac{6x}{6x} = 1$
Answer: $6x(2x^2 - 3x + 1)$
Example 3: Factor $15a^3b^2 + 10a^2b - 25a^2b^2$
GCF = $5a^2b$
$= 5a^2b(3ab + 2 - 5b)$
Answer: $5a^2b(3ab + 2 - 5b)$
GCF = $5a^2b$
$= 5a^2b(3ab + 2 - 5b)$
Answer: $5a^2b(3ab + 2 - 5b)$
Always Factor Out GCF FIRST!
This makes all other factoring methods easier
This makes all other factoring methods easier
4. Factor Quadratics Using Algebra Tiles
Algebra Tiles: Visual models to represent factoring
• Large square: $x^2$
• Rectangle: $x$
• Small square: $1$
• Large square: $x^2$
• Rectangle: $x$
• Small square: $1$
Using Tiles to Factor $x^2 + 5x + 6$:
1. Place 1 large square ($x^2$)
2. Place 5 rectangles ($5x$)
3. Place 6 small squares ($6$)
4. Arrange into rectangle
5. Dimensions give factors: $(x + 2)(x + 3)$
The rectangle's dimensions are the factors!
1. Place 1 large square ($x^2$)
2. Place 5 rectangles ($5x$)
3. Place 6 small squares ($6$)
4. Arrange into rectangle
5. Dimensions give factors: $(x + 2)(x + 3)$
The rectangle's dimensions are the factors!
5. Factor Quadratics with Leading Coefficient 1
Form: $x^2 + bx + c$ where coefficient of $x^2$ is 1
Factoring Pattern:
$$x^2 + bx + c = (x + m)(x + n)$$
where:
• $m + n = b$ (sum equals middle coefficient)
• $m \times n = c$ (product equals constant term)
Find two numbers that multiply to $c$ and add to $b$
$$x^2 + bx + c = (x + m)(x + n)$$
where:
• $m + n = b$ (sum equals middle coefficient)
• $m \times n = c$ (product equals constant term)
Find two numbers that multiply to $c$ and add to $b$
Steps to Factor $x^2 + bx + c$:
Step 1: Find two numbers whose product = $c$
Step 2: Check if their sum = $b$
Step 3: Write as $(x + m)(x + n)$
Step 4: Verify by multiplying (FOIL)
Step 1: Find two numbers whose product = $c$
Step 2: Check if their sum = $b$
Step 3: Write as $(x + m)(x + n)$
Step 4: Verify by multiplying (FOIL)
Example 1: Factor $x^2 + 7x + 12$
Need: Two numbers that multiply to 12 and add to 7
Factors of 12:
• 1 × 12 = 12, and 1 + 12 = 13 ✗
• 2 × 6 = 12, and 2 + 6 = 8 ✗
• 3 × 4 = 12, and 3 + 4 = 7 ✓
Answer: $(x + 3)(x + 4)$
Check: $(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$ ✓
Need: Two numbers that multiply to 12 and add to 7
Factors of 12:
• 1 × 12 = 12, and 1 + 12 = 13 ✗
• 2 × 6 = 12, and 2 + 6 = 8 ✗
• 3 × 4 = 12, and 3 + 4 = 7 ✓
Answer: $(x + 3)(x + 4)$
Check: $(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$ ✓
Example 2: Factor $x^2 - 5x + 6$
Need: Product = 6, Sum = -5
Both numbers must be negative!
• (-1) × (-6) = 6, (-1) + (-6) = -7 ✗
• (-2) × (-3) = 6, (-2) + (-3) = -5 ✓
Answer: $(x - 2)(x - 3)$
Need: Product = 6, Sum = -5
Both numbers must be negative!
• (-1) × (-6) = 6, (-1) + (-6) = -7 ✗
• (-2) × (-3) = 6, (-2) + (-3) = -5 ✓
Answer: $(x - 2)(x - 3)$
Example 3: Factor $x^2 + 2x - 15$
Need: Product = -15, Sum = 2
One positive, one negative!
• 5 × (-3) = -15, 5 + (-3) = 2 ✓
Answer: $(x + 5)(x - 3)$
Need: Product = -15, Sum = 2
One positive, one negative!
• 5 × (-3) = -15, 5 + (-3) = 2 ✓
Answer: $(x + 5)(x - 3)$
Example 4: Factor $x^2 - x - 20$
Product = -20, Sum = -1
• 4 × (-5) = -20, 4 + (-5) = -1 ✓
Answer: $(x + 4)(x - 5)$
Product = -20, Sum = -1
• 4 × (-5) = -20, 4 + (-5) = -1 ✓
Answer: $(x + 4)(x - 5)$
Sign Rules:
• Both positive: $c$ is positive, $b$ is positive
• Both negative: $c$ is positive, $b$ is negative
• Different signs: $c$ is negative
• Both positive: $c$ is positive, $b$ is positive
• Both negative: $c$ is positive, $b$ is negative
• Different signs: $c$ is negative
6. Factor Quadratics with Other Leading Coefficients
Form: $ax^2 + bx + c$ where $a \neq 1$
Methods: AC Method (Splitting the Middle Term) or Trial and Error
Methods: AC Method (Splitting the Middle Term) or Trial and Error
AC Method (Product-Sum Method)
AC Method Steps:
Step 1: Multiply $a \times c$ to get AC
Step 2: Find two numbers that multiply to AC and add to $b$
Step 3: Split middle term using these two numbers
Step 4: Factor by grouping
Step 1: Multiply $a \times c$ to get AC
Step 2: Find two numbers that multiply to AC and add to $b$
Step 3: Split middle term using these two numbers
Step 4: Factor by grouping
Example 1: Factor $2x^2 + 7x + 3$
Step 1: $AC = 2 \times 3 = 6$
Step 2: Need two numbers: product = 6, sum = 7
• 1 × 6 = 6, 1 + 6 = 7 ✓
Step 3: Split middle term:
$2x^2 + 1x + 6x + 3$
Step 4: Factor by grouping:
$= x(2x + 1) + 3(2x + 1)$
$= (2x + 1)(x + 3)$
Answer: $(2x + 1)(x + 3)$
Step 1: $AC = 2 \times 3 = 6$
Step 2: Need two numbers: product = 6, sum = 7
• 1 × 6 = 6, 1 + 6 = 7 ✓
Step 3: Split middle term:
$2x^2 + 1x + 6x + 3$
Step 4: Factor by grouping:
$= x(2x + 1) + 3(2x + 1)$
$= (2x + 1)(x + 3)$
Answer: $(2x + 1)(x + 3)$
Example 2: Factor $3x^2 + 10x + 8$
$AC = 3 \times 8 = 24$
Need: product = 24, sum = 10
• 4 × 6 = 24, 4 + 6 = 10 ✓
Split: $3x^2 + 4x + 6x + 8$
Group: $x(3x + 4) + 2(3x + 4)$
$= (3x + 4)(x + 2)$
Answer: $(3x + 4)(x + 2)$
$AC = 3 \times 8 = 24$
Need: product = 24, sum = 10
• 4 × 6 = 24, 4 + 6 = 10 ✓
Split: $3x^2 + 4x + 6x + 8$
Group: $x(3x + 4) + 2(3x + 4)$
$= (3x + 4)(x + 2)$
Answer: $(3x + 4)(x + 2)$
Example 3: Factor $6x^2 - 13x + 6$
$AC = 6 \times 6 = 36$
Need: product = 36, sum = -13
• (-4) × (-9) = 36, (-4) + (-9) = -13 ✓
Split: $6x^2 - 4x - 9x + 6$
Group: $2x(3x - 2) - 3(3x - 2)$
$= (3x - 2)(2x - 3)$
Answer: $(3x - 2)(2x - 3)$
$AC = 6 \times 6 = 36$
Need: product = 36, sum = -13
• (-4) × (-9) = 36, (-4) + (-9) = -13 ✓
Split: $6x^2 - 4x - 9x + 6$
Group: $2x(3x - 2) - 3(3x - 2)$
$= (3x - 2)(2x - 3)$
Answer: $(3x - 2)(2x - 3)$
7. Factor Quadratics: Special Cases
Special factoring patterns that appear frequently
Difference of Squares
Difference of Squares Pattern:
$$a^2 - b^2 = (a + b)(a - b)$$
Recognition:
• Two perfect squares
• Separated by subtraction (minus)
• No middle term
$$a^2 - b^2 = (a + b)(a - b)$$
Recognition:
• Two perfect squares
• Separated by subtraction (minus)
• No middle term
Example 1: Factor $x^2 - 25$
$x^2 - 25 = x^2 - 5^2$
$= (x + 5)(x - 5)$
Answer: $(x + 5)(x - 5)$
$x^2 - 25 = x^2 - 5^2$
$= (x + 5)(x - 5)$
Answer: $(x + 5)(x - 5)$
Example 2: Factor $4x^2 - 49$
$4x^2 - 49 = (2x)^2 - 7^2$
$= (2x + 7)(2x - 7)$
Answer: $(2x + 7)(2x - 7)$
$4x^2 - 49 = (2x)^2 - 7^2$
$= (2x + 7)(2x - 7)$
Answer: $(2x + 7)(2x - 7)$
Example 3: Factor $9a^2 - 16b^2$
$= (3a)^2 - (4b)^2$
$= (3a + 4b)(3a - 4b)$
Answer: $(3a + 4b)(3a - 4b)$
$= (3a)^2 - (4b)^2$
$= (3a + 4b)(3a - 4b)$
Answer: $(3a + 4b)(3a - 4b)$
Perfect Square Trinomials
Perfect Square Patterns:
1. Square of a Sum:
$$a^2 + 2ab + b^2 = (a + b)^2$$
2. Square of a Difference:
$$a^2 - 2ab + b^2 = (a - b)^2$$
Recognition:
• First and last terms are perfect squares
• Middle term = $2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}$
1. Square of a Sum:
$$a^2 + 2ab + b^2 = (a + b)^2$$
2. Square of a Difference:
$$a^2 - 2ab + b^2 = (a - b)^2$$
Recognition:
• First and last terms are perfect squares
• Middle term = $2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}$
Example 4: Factor $x^2 + 6x + 9$
Check if perfect square:
• $x^2$ is perfect square: $x$
• $9$ is perfect square: $3$
• Middle: $2 \times x \times 3 = 6x$ ✓
$x^2 + 6x + 9 = (x + 3)^2$
Answer: $(x + 3)^2$
Check if perfect square:
• $x^2$ is perfect square: $x$
• $9$ is perfect square: $3$
• Middle: $2 \times x \times 3 = 6x$ ✓
$x^2 + 6x + 9 = (x + 3)^2$
Answer: $(x + 3)^2$
Example 5: Factor $4x^2 - 12x + 9$
• $(2x)^2 = 4x^2$ ✓
• $3^2 = 9$ ✓
• $2 \times 2x \times 3 = 12x$ ✓
$= (2x - 3)^2$
Answer: $(2x - 3)^2$
• $(2x)^2 = 4x^2$ ✓
• $3^2 = 9$ ✓
• $2 \times 2x \times 3 = 12x$ ✓
$= (2x - 3)^2$
Answer: $(2x - 3)^2$
Example 6: Factor $9x^2 + 30x + 25$
$(3x)^2 + 2(3x)(5) + 5^2 = (3x + 5)^2$
Answer: $(3x + 5)^2$
$(3x)^2 + 2(3x)(5) + 5^2 = (3x + 5)^2$
Answer: $(3x + 5)^2$
8. Factor by Grouping
Factor by Grouping: Method for factoring polynomials with 4 or more terms
Steps for Factoring by Grouping:
Step 1: Group terms in pairs (usually first two and last two)
Step 2: Factor out GCF from each pair
Step 3: Factor out common binomial
Step 4: Check by multiplying
Step 1: Group terms in pairs (usually first two and last two)
Step 2: Factor out GCF from each pair
Step 3: Factor out common binomial
Step 4: Check by multiplying
Example 1: Factor $x^3 + 3x^2 + 2x + 6$
Step 1: Group:
$(x^3 + 3x^2) + (2x + 6)$
Step 2: Factor each group:
$x^2(x + 3) + 2(x + 3)$
Step 3: Factor out $(x + 3)$:
$(x + 3)(x^2 + 2)$
Answer: $(x + 3)(x^2 + 2)$
Step 1: Group:
$(x^3 + 3x^2) + (2x + 6)$
Step 2: Factor each group:
$x^2(x + 3) + 2(x + 3)$
Step 3: Factor out $(x + 3)$:
$(x + 3)(x^2 + 2)$
Answer: $(x + 3)(x^2 + 2)$
Example 2: Factor $6x^2 + 9x + 4x + 6$
Group: $(6x^2 + 9x) + (4x + 6)$
Factor: $3x(2x + 3) + 2(2x + 3)$
Result: $(2x + 3)(3x + 2)$
Answer: $(2x + 3)(3x + 2)$
Group: $(6x^2 + 9x) + (4x + 6)$
Factor: $3x(2x + 3) + 2(2x + 3)$
Result: $(2x + 3)(3x + 2)$
Answer: $(2x + 3)(3x + 2)$
Example 3: Factor $xy + 3x + 2y + 6$
$(xy + 3x) + (2y + 6)$
$= x(y + 3) + 2(y + 3)$
$= (y + 3)(x + 2)$
Answer: $(y + 3)(x + 2)$
$(xy + 3x) + (2y + 6)$
$= x(y + 3) + 2(y + 3)$
$= (y + 3)(x + 2)$
Answer: $(y + 3)(x + 2)$
Important: If grouping doesn't work at first, try rearranging terms!
9. Factor Using a Quadratic Pattern
Quadratic Pattern: Expressions that look like quadratics but use different variables or powers
Example 1: Factor $x^4 - 5x^2 + 6$
Substitute: Let $u = x^2$
Then: $u^2 - 5u + 6$
Factor: $(u - 2)(u - 3)$
Substitute back:
$(x^2 - 2)(x^2 - 3)$
Answer: $(x^2 - 2)(x^2 - 3)$
Substitute: Let $u = x^2$
Then: $u^2 - 5u + 6$
Factor: $(u - 2)(u - 3)$
Substitute back:
$(x^2 - 2)(x^2 - 3)$
Answer: $(x^2 - 2)(x^2 - 3)$
Example 2: Factor $(x + 1)^2 + 5(x + 1) + 6$
Let $u = (x + 1)$
$u^2 + 5u + 6 = (u + 2)(u + 3)$
Substitute back:
$[(x + 1) + 2][(x + 1) + 3]$
$= (x + 3)(x + 4)$
Answer: $(x + 3)(x + 4)$
Let $u = (x + 1)$
$u^2 + 5u + 6 = (u + 2)(u + 3)$
Substitute back:
$[(x + 1) + 2][(x + 1) + 3]$
$= (x + 3)(x + 4)$
Answer: $(x + 3)(x + 4)$
Example 3: Factor $x^6 - 9$
Recognize difference of squares:
$(x^3)^2 - 3^2$
$= (x^3 + 3)(x^3 - 3)$
Answer: $(x^3 + 3)(x^3 - 3)$
Recognize difference of squares:
$(x^3)^2 - 3^2$
$= (x^3 + 3)(x^3 - 3)$
Answer: $(x^3 + 3)(x^3 - 3)$
10. Factor Polynomials (Complete Strategy)
Complete Factoring Strategy (Always Use This Order!):
Step 1: GCF
Always factor out the GCF first!
Step 2: Count Terms
• 2 terms → Check for difference of squares
• 3 terms → Check for perfect square or factor as quadratic
• 4+ terms → Try factoring by grouping
Step 3: Special Patterns
• Difference of squares: $a^2 - b^2$
• Perfect square trinomial: $a^2 \pm 2ab + b^2$
Step 4: Factor Completely
Check if any factors can be factored further
Step 5: Check
Multiply to verify
Step 1: GCF
Always factor out the GCF first!
Step 2: Count Terms
• 2 terms → Check for difference of squares
• 3 terms → Check for perfect square or factor as quadratic
• 4+ terms → Try factoring by grouping
Step 3: Special Patterns
• Difference of squares: $a^2 - b^2$
• Perfect square trinomial: $a^2 \pm 2ab + b^2$
Step 4: Factor Completely
Check if any factors can be factored further
Step 5: Check
Multiply to verify
Example 1: Factor $3x^3 - 27x$
Step 1: GCF
$= 3x(x^2 - 9)$
Step 2: Check $x^2 - 9$
Difference of squares!
$= 3x(x + 3)(x - 3)$
Answer: $3x(x + 3)(x - 3)$
Step 1: GCF
$= 3x(x^2 - 9)$
Step 2: Check $x^2 - 9$
Difference of squares!
$= 3x(x + 3)(x - 3)$
Answer: $3x(x + 3)(x - 3)$
Example 2: Factor $4x^2 - 16x + 16$
GCF: 4
$= 4(x^2 - 4x + 4)$
Perfect square trinomial:
$= 4(x - 2)^2$
Answer: $4(x - 2)^2$
GCF: 4
$= 4(x^2 - 4x + 4)$
Perfect square trinomial:
$= 4(x - 2)^2$
Answer: $4(x - 2)^2$
Example 3: Factor $2x^3 + 6x^2 + 2x + 6$
GCF: 2
$= 2(x^3 + 3x^2 + x + 3)$
Factor by grouping:
$= 2[x^2(x + 3) + 1(x + 3)]$
$= 2(x + 3)(x^2 + 1)$
Answer: $2(x + 3)(x^2 + 1)$
GCF: 2
$= 2(x^3 + 3x^2 + x + 3)$
Factor by grouping:
$= 2[x^2(x + 3) + 1(x + 3)]$
$= 2(x + 3)(x^2 + 1)$
Answer: $2(x + 3)(x^2 + 1)$
Example 4: Factor $x^4 - 16$
Difference of squares:
$= (x^2)^2 - 4^2$
$= (x^2 + 4)(x^2 - 4)$
Factor further: $x^2 - 4$ is also difference of squares!
$= (x^2 + 4)(x + 2)(x - 2)$
Answer: $(x^2 + 4)(x + 2)(x - 2)$
Difference of squares:
$= (x^2)^2 - 4^2$
$= (x^2 + 4)(x^2 - 4)$
Factor further: $x^2 - 4$ is also difference of squares!
$= (x^2 + 4)(x + 2)(x - 2)$
Answer: $(x^2 + 4)(x + 2)(x - 2)$
Example 5: Factor $6x^2 + 13x - 5$
No GCF
Use AC method:
$AC = 6 \times (-5) = -30$
Need: product = -30, sum = 13
• 15 × (-2) = -30, 15 + (-2) = 13 ✓
Split: $6x^2 + 15x - 2x - 5$
Group: $3x(2x + 5) - 1(2x + 5)$
$= (2x + 5)(3x - 1)$
Answer: $(2x + 5)(3x - 1)$
No GCF
Use AC method:
$AC = 6 \times (-5) = -30$
Need: product = -30, sum = 13
• 15 × (-2) = -30, 15 + (-2) = 13 ✓
Split: $6x^2 + 15x - 2x - 5$
Group: $3x(2x + 5) - 1(2x + 5)$
$= (2x + 5)(3x - 1)$
Answer: $(2x + 5)(3x - 1)$
Summary: Factoring Methods
| Method | When to Use | Pattern | Example |
|---|---|---|---|
| GCF | Always try first | $ab + ac = a(b + c)$ | $6x^2 + 9x = 3x(2x + 3)$ |
| Difference of Squares | Two perfect squares with minus | $a^2 - b^2 = (a+b)(a-b)$ | $x^2 - 25 = (x+5)(x-5)$ |
| Perfect Square Trinomial | First/last perfect squares, middle = $2ab$ | $a^2 \pm 2ab + b^2 = (a \pm b)^2$ | $x^2 + 6x + 9 = (x+3)^2$ |
| Simple Trinomial | $x^2 + bx + c$ | Find two numbers: sum = $b$, product = $c$ | $x^2 + 5x + 6 = (x+2)(x+3)$ |
| AC Method | $ax^2 + bx + c$ where $a \neq 1$ | Find numbers: sum = $b$, product = $ac$ | $2x^2 + 7x + 3 = (2x+1)(x+3)$ |
| Grouping | 4 or more terms | Group pairs, factor each, factor common binomial | $x^3 + 2x^2 + 3x + 6 = (x+2)(x^2+3)$ |
Quick Reference: Special Patterns
| Pattern Name | Factored Form | Expanded Form |
|---|---|---|
| Difference of Squares | $(a + b)(a - b)$ | $a^2 - b^2$ |
| Perfect Square (Sum) | $(a + b)^2$ | $a^2 + 2ab + b^2$ |
| Perfect Square (Difference) | $(a - b)^2$ | $a^2 - 2ab + b^2$ |
| Sum of Squares | Cannot factor (with real numbers) | $a^2 + b^2$ |
Factoring Flowchart
Start Here:
1. Factor out GCF?
→ Yes: Factor it out and continue
→ No: Continue to next step
2. How many terms?
Two Terms:
• Is it $a^2 - b^2$? → Difference of squares
• Is it $a^2 + b^2$? → Cannot factor
Three Terms:
• Is it a perfect square trinomial? → $(a \pm b)^2$
• Is it $x^2 + bx + c$? → Find two numbers
• Is it $ax^2 + bx + c$? → Use AC method
Four or More Terms:
• Try grouping in pairs
3. Can any factors be factored further?
→ Yes: Factor completely
→ No: Done!
4. Check by multiplying!
1. Factor out GCF?
→ Yes: Factor it out and continue
→ No: Continue to next step
2. How many terms?
Two Terms:
• Is it $a^2 - b^2$? → Difference of squares
• Is it $a^2 + b^2$? → Cannot factor
Three Terms:
• Is it a perfect square trinomial? → $(a \pm b)^2$
• Is it $x^2 + bx + c$? → Find two numbers
• Is it $ax^2 + bx + c$? → Use AC method
Four or More Terms:
• Try grouping in pairs
3. Can any factors be factored further?
→ Yes: Factor completely
→ No: Done!
4. Check by multiplying!
Success Tips for Factoring Polynomials:
✓ ALWAYS factor out GCF first
✓ Memorize special patterns (difference of squares, perfect squares)
✓ For simple trinomials: find two numbers (sum and product)
✓ For complex trinomials: use AC method
✓ Factor completely - check if factors can be factored further
✓ Always check your answer by multiplying back
✓ GCF: take LOWEST power; LCM: take HIGHEST power
✓ Practice recognizing patterns
✓ Draw algebra tiles to visualize
✓ When in doubt, multiply it out to check!
✓ ALWAYS factor out GCF first
✓ Memorize special patterns (difference of squares, perfect squares)
✓ For simple trinomials: find two numbers (sum and product)
✓ For complex trinomials: use AC method
✓ Factor completely - check if factors can be factored further
✓ Always check your answer by multiplying back
✓ GCF: take LOWEST power; LCM: take HIGHEST power
✓ Practice recognizing patterns
✓ Draw algebra tiles to visualize
✓ When in doubt, multiply it out to check!