Factoring Trinomials and Polynomials: College Algebra Study Notes

Polynomials and Factoring

Introduction to Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. Understanding how to simplify and factor polynomials is a foundational skill in college algebra.

• Polynomial: An expression of the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$ where $a_i$ are constants.

• Degree: The highest power of the variable in the polynomial.

• Monomial, Binomial, Trinomial: Polynomials with one, two, or three terms, respectively.

Simplifying Polynomials

Simplifying polynomials involves combining like terms and performing basic arithmetic operations.

• Example: $(x+1)(x+5)$

  • Expand using distributive property (FOIL):

  • $x \cdot x + x \cdot 5 + 1 \cdot x + 1 \cdot 5 = x^2 + 5x + x + 5 = x^2 + 6x + 5$

• Example: $(m-4)(m+6)$

  • Expand: $m^2 + 6m - 4m - 24 = m^2 + 2m - 24$

• Example: $(k-7)(k-3)$

  • Expand: $k^2 - 3k - 7k + 21 = k^2 - 10k + 21$

Factoring Trinomials

Factoring Trinomials of the Form $x^2 + bx + c$

Factoring trinomials is the process of expressing a quadratic polynomial as a product of two binomials. This is a key skill for solving quadratic equations and simplifying expressions.

• To factor $x^2 + bx + c$, find two integers that multiply to c and add to b.

• Write the trinomial as $(x + p)(x + q)$, where $p$ and $q$ are the integers found.

Example: Factor $x^2 + 7x + 12$

• Find two numbers that multiply to 12 and add to 7: 3 and 4.

• So, $x^2 + 7x + 12 = (x + 3)(x + 4)$

Factoring Practice Sets

• Set 1 (All positive coefficients):

  • $n^2 + 9n + 20 = (n + 4)(n + 5)$

  • $w^2 + 9w + 18 = (w + 6)(w + 3)$

  • $a^2 + 10a + 24 = (a + 6)(a + 4)$

  • $y^2 + 2y + 1 = (y + 1)(y + 1)$

• Set 2 (Mixed signs):

  • $n^2 + 3n - 18 = (n + 6)(n - 3)$

  • $x^2 + 2x - 8 = (x + 4)(x - 2)$

  • $g^2 + g - 6 = (g + 3)(g - 2)$

  • $c^2 + 4c - 45 = (c + 9)(c - 5)$

• Set 3 (Negative coefficients):

  • $a^2 - 2a - 3 = (a - 3)(a + 1)$

  • $x^2 - 7x - 30 = (x - 10)(x + 3)$

  • $b^2 - 2b - 63 = (b - 9)(b + 7)$

  • $k^2 - 12k - 64 = (k - 16)(k + 4)$

• Set 4 (Positive constant term):

  • $x^2 - 11x + 24 = (x - 8)(x - 3)$

  • $x^2 - 14x + 49 = (x - 7)(x - 7)$

  • $y^2 - 17y + 72 = (y - 9)(y - 8)$

  • $m^2 - 15m + 50 = (m - 10)(m - 5)$

Mixed Practice

• $a^2 + 6a - 16 = (a + 8)(a - 2)$

• $x^2 - 14x - 72 = (x - 18)(x + 4)$

• $y^2 + 13y + 40 = (y + 8)(y + 5)$

• $w^2 - 16w + 48 = (w - 12)(w - 4)$

• $n^2 - n - 30 = (n - 6)(n + 5)$

• $k^2 + 13k + 42 = (k + 7)(k + 6)$

Factoring with a Greatest Common Factor (GCF)

Factoring Out the GCF

Before factoring a trinomial, always check for a Greatest Common Factor (GCF) among all terms. Factoring out the GCF simplifies the polynomial and makes further factoring easier.

• GCF: The largest integer or variable expression that divides each term of the polynomial.

• Example: $4k^2 + 12k + 8$

  • GCF is 4: $4(k^2 + 3k + 2)$

  • Factor the trinomial: $4(k + 1)(k + 2)$

• Example: $2x^2 - 8x - 24$

  • GCF is 2: $2(x^2 - 4x - 12)$

  • Factor the trinomial: $2(x - 6)(x + 2)$

Factoring Higher Degree Polynomials

Some polynomials may have degrees higher than 2 or include multiple variables. The process is similar: factor out the GCF, then factor the remaining polynomial if possible.

• Example: $3a^3 + 30a^2 + 63a$

  • GCF is 3a: $3a(a^2 + 10a + 21)$

  • Factor the trinomial: $3a(a + 7)(a + 3)$

• Example: $5x^2y - 15xy - 140y$

  • GCF is 5y: $5y(x^2 - 3x - 28)$

  • Factor the trinomial: $5y(x - 7)(x + 4)$

Checking Factoring by FOIL

FOIL Method

The FOIL method (First, Outer, Inner, Last) is used to multiply two binomials and check the correctness of factoring.

• Example: $(x + 3)(x + 4)$

  • First: $x \cdot x = x^2$

  • Outer: $x \cdot 4 = 4x$

  • Inner: $3 \cdot x = 3x$

  • Last: $3 \cdot 4 = 12$

  • Sum: $x^2 + 4x + 3x + 12 = x^2 + 7x + 12$

Summary Table: Factoring Trinomials

Trinomial	Factored Form	GCF (if any)
$x^2 + 5x + 6$	$(x + 2)(x + 3)$	None
$4k^2 + 12k + 8$	$4(k + 1)(k + 2)$	4
$3a^3 + 30a^2 + 63a$	$3a(a + 7)(a + 3)$	$3a$
$x^2 - 14x - 72$	$(x - 18)(x + 4)$	None
$5x^2y - 15xy - 140y$	$5y(x - 7)(x + 4)$	$5y$

Visual Example: Factoring Trinomials

Trinomials like $x^2 + bx + c$ can be factored back into a product of binomials. This process is essential for solving quadratic equations and simplifying algebraic expressions.

Additional info: The notes above expand on the brief worksheet instructions, providing definitions, step-by-step examples, and a summary table for reference. This guide is suitable for college algebra students preparing for exams on polynomial factoring.