Factoring Polynomials: Methods and Techniques

Posted on Nov 22, 2024 in Mathematics

Topic: Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors. Several factorization methods exist, each applying to specific polynomial forms.

Factoring Methods

• Greatest Common Factor
• Difference of Squares
• Quadratic Trinomials: x² + bx + c
• Quadratic Trinomials: ax² + bx + c
• Sum or Difference of Cubes
• Factoring by Grouping

A. Greatest Common Factor

Important Definitions

Numerical Factors

Factors of a number are all the numbers that, when multiplied together, result in that number.

Example: Factors of 24:
1 x 24 = 24
2 x 12 = 24
3 x 8 = 24
4 x 6 = 24

Factors of a Term

A factor is each number or variable within a term.

Example: In the term -4x⁴y⁹z², the factors are -4, x⁴, y⁹, and z².

There are two types of factors in a term:

1. The numerical factor (a number) is called the coefficient.
2. The literal factors are the variables (x⁴, y⁹, z²).

Example: The coefficient of -4x⁴y⁹z² is -4.

Greatest Common Factor (GCF)

The GCF of terms in a polynomial includes:

• The greatest common factor of the numerical coefficients.
• The lowest power of any common variables.

Examples:

• y + 7 = Not factorable (no common factors)
• 5x – 10 = 5(x – 2)
• p⁵ + 7p³ – 9p² = p²(p³ + 7p – 9)
• 12x⁵ – 18x⁸ + 24x¹⁰ = 6x⁵(2 – 3x³ + 4x⁵)
• 36x⁷y³ – 27x⁶y⁵ + 81x⁴y⁸ = 9x⁴y³(4x³ – 3x²y² + 9y⁵)
• (x + 3)³ + (x + 3)² = (x + 3)²[(x + 3) + 1]
• 35(p – 6)⁸ – 50(p – 6)¹² = 5(p – 6)⁸[7 – 10(p – 6)⁴]
• 48a⁷b²(w + z)⁶ – 36a⁶b⁵(w + z)⁴ = 12a⁶b²(w + z)⁴[4a(w + z)² – 3b³]

Practice: Factor if possible.

• 6x + 3y =
• 12x + 8y =
• 6x² – 14x =
• 15x² – 6x =
• 28y² + 4y =
• 42y² + 6y =
• 20xy – 15x =
• 12x³ + 10x² =
• 18a^-4b⁷ – 27a³b⁸ =
• 42y⁶z⁵ + 21y¹¹z⁹ =
• 6z + 9z² – 15z³ =
• 63w¹⁵ – 36w⁹ + 81w⁵ =
• 18x⁵y⁹z² – 27x³y⁷z⁴ + 45x²y⁶z⁸ =
• 36p²q⁸z⁹ + 18p⁴q⁷z³ – 24p¹³q¹⁰z⁴ =
• 5(a – 6)³ – (a – 6) =
• w³(w + 3)⁵ + w⁷(w + 3)⁶ =
• t(t + 1)⁷ + 4(t + 1)¹² =
• 80m³(t – 4)⁸ + 18m⁵(t – 4)⁶ =
• 3(x – z) – z(z – x) =
• p(c – 7) + 5(7 – c) =

B. Difference of Squares

This factorization method applies to a binomial that is a difference of two perfect squares.

Example: Finding squares:
(4)² = 16
(-9)² = 81
(r)² = r²
(t⁴)² = t⁸
(-7p⁸)² = 49p¹⁶

Definition: Conjugate binomials differ only in the sign of the second term.

Examples: Conjugates:
(w – 4) → (w + 4)
(x + 5) → (x – 5)
(2p + z) → (2p – z)

Important Note: Multiplying conjugate binomials results in a difference of squares. Factoring a difference of squares results in conjugate binomials.

Examples:

• p² – 4 = (p + 2)(p – 2)
• 16 – m² = (4 – m)(4 + m)
• 25t² – 64q² = (5t + 8q)(5t – 8q)
• 144x⁶ – 1 = (12x³ + 1)(12x³ – 1)
• y⁴ – 1 = (y² + 1)(y² – 1) = (y² + 1)(y + 1)(y – 1)

Practice: Factor.

• t² – 1 =
• w² – 100 =
• y⁸ – 64 =
• q² – 49 =
• 4b⁶ – 9 =
• 1 + f² =
• 4t² – y² =
• d² – 36g² =
• 81 – h¹⁰ =
• 9 – 25c² =
• k⁴ – 64j² =
• p² + q² =
• 144 – m¹⁴ =
• 49n² – 121 =
• b² – 4c² =
• 25g⁸ – 144h²⁰ =
• 36 – 121s¹⁶ =
• 49 + y⁴ =
• y² – z⁴ =
• 64d² – 9k¹⁴ =
• 169 – b¹⁰ =
• 100 + 36p² =
• (3 + u)⁶ – (t + 4)¹⁰ =
• (w – 4)² – (t + 9)² =
• (d – 8)⁸ – (c – 6)¹⁰ =

C. Factoring Quadratic Trinomials: x² + bx + c

Factoring is the reverse of multiplication. To factor a quadratic trinomial, find two binomials that multiply to give x² + bx + c.

Example: (x + 3)(x + 5) = x² + 8x + 15

Procedure:

1. Find all factors of c.
2. If c is positive, choose factors that add up to b. Both factors have the same sign as b.
3. If c is negative, choose factors that subtract to give |b|. The larger factor has the same sign as b.

Examples:

• x² + 7x + 12 = (x + 3)(x + 4)
• x² – 9x + 18 = (x – 3)(x – 6)
• y² + 6y – 27 = (y + 9)(y – 3)
• z² – 4z – 45 = (z – 9)(z + 5)

Practice: Factor.

• w² – w – 2 =
• c² – 10c + 25 =
• t² + 8t + 12 =
• x² – 4x – 12 =
• y² + 8y – 9 =
• d² + 10d + 24 =
• x² + 12x + 20 =
• a² – a – 30 =
• y² + y – 72 =
• y² + 12y + 36 =
• w² – w – 12 =
• d² + 9d – 10 =
• r² – 15r + 54 =
• f² + 7f + 12 =
• y² + 12y – 45 =
• x² + 9x + 20 =
• w² – 11w + 28 =
• q² + 5q – 14 =
• s² – 8s + 12 =
• z² – 3z – 54 =
• a² + 8a – 24 =
• a² – 14xy – 63y² =
• a² – ab – 56b² =
• y² + 21y + 98 =
• x² + 9x + 12 =
• 6 + 5x – x² =
• c² – 9c – 10 =
• d² + 3d – 54 =
• x⁴ – 5x² – 36 =
• t⁴ + 6t² + 9 =

D. Quadratic Trinomials: ax² + bx + c

Use similar criteria as factoring x² + bx + c.

Procedure:

ax² + bx + c = ( _ x + _ )( _ x + _ )

• The product of the first terms must be a.
• The sum of the outer and inner products must be b.
• The product of the last terms must be c.

Example:

5x² + 11x + 2 = (5x + 1)(x + 2)

Practice: Factor.

Many examples and practice problems were provided in the original document. These have been omitted for brevity.

E. Factoring Sums or Differences of Cubes

This method is used for binomials that are sums or differences of two perfect cubes.

a³ + b³ = (a + b)(a² – ab + b²)
a³ – b³ = (a – b)(a² + ab + b²)

Important Note:

• To find the cube root of a number, find the number that, when multiplied by itself three times, equals the given number.
• To find the cube root of a variable with an exponent, divide the exponent by 3.

Examples:

• x³ + 8 = (x + 2)(x² – 2x + 4)
• 27 – y⁶ = (3 – y²)(9 + 3y² + y⁴)
• 8p⁹ + 64 = (2p³ + 4)(4p⁶ – 8p³ + 16)

Practice: Factor.

Many examples and practice problems were provided in the original document. These have been omitted for brevity.

F. Factoring by Grouping

This method is used when other methods are not applicable, typically for polynomials with four terms.

Procedure:

1. Group terms with common factors using the associative property.
2. Factor each resulting binomial.
3. Factor again using the GCF. The terms in the parentheses must be identical.

Examples:

• x³ + 2x² + 3x + 6 = (x + 2)(x² + 3)
• 3x³ + 6x² + 2x + 4 = (x + 2)(3x² + 2)
• 4p⁴ – 12p² – 3p² + 9 = (p² – 3)(4p² – 3)

Practice: Factor.

Many examples and practice problems were provided in the original document. These have been omitted for brevity.