Factoring and Rational Expressions – Study Notes
Factoring and Rational Expressions – Study Notes
Purpose of factoring: rewrite polynomials as products of lower-degree polynomials to simplify, solve, or recognize special forms. The lecture emphasizes identifying forms quickly and then applying the appropriate factoring strategy.
1) Recognizing and Factoring Special Product Forms
Difference of squares
Formula: a^2 - b^2 = (a-b)(a+b)
Example from lecture: if you have something that is a difference of two squares, factor into a product of conjugates.
Square-like (perfect square) trinomials
If an expression can be written as A^2 + 2AB + B^2, it factors as (A+B)^2.
If it can be written as A^2 - 2AB + B^2, it factors as (A-B)^2.
Strategy shown: express the first term as a square, attach the cross-term, and recognize a square of a binomial.
Difference of cubes
Formula: a^3 - b^3 = (a-b)(a^2 + ab + b^2)
The lecture illustrated identifying a cube in the expression, writing it as a cube, and applying the difference-of-cubes formula.
Sum of cubes
Formula: a^3 + b^3 = (a+b)(a^2 - ab + b^2)
The lecture also notes this can be used when the expression is a sum of two cubes.
Difference of squares vs. difference of cubes vs. sum of cubes in practice
Some problems yield simple factoring by recognizing the form immediately (e.g., a quadratic that is a difference of squares).
Others require rewriting terms to fit a known pattern (AC method or grouping may be needed for quadratics or higher-degree polynomials).
Quick check by expansion
After guessing a factoring form, expand to ensure you recover the original polynomial.
This helps verify that the chosen a and b (in forms like $(a+b)^2$) or the selected factors are correct.
2) Quadratic Case and the AC Method
AC method (for ax^2 + bx + c with a ≠ 1)
Goal: rewrite middle term using two numbers that multiply to $ac$ and add to $b$.
Steps:
1) Compute $ac$.
2) Find two numbers $m$ and $n$ such that m+n=b \,\text{ and } \, mn = ac.
3) Rewrite the quadratic as ax^2 + mx + nx + c.
4) Factor by grouping to obtain a binomial factor times another binomial.This method is a structured alternative to factoring by inspection when $a
eq 1$.
Recognizing perfect-square structure inside quadratics
If you can express a quadratic as a square of a binomial after grouping, you may factor quickly as a square.
The lecture shows using this approach to identify a and b in a form like (\text{something})^2 = a^2 + 2ab + b^2.
3) Factoring by Grouping (four-term polynomials)
When a polynomial has four terms, a common strategy is grouping:
Group the first two terms and the last two terms separately.
Factor out the greatest common factor (GCF) from each group.
If a common binomial factor emerges, factor it out to finish.
Example pattern mentioned:
Break into two pairs to reveal a common binomial factor, which then allows a final factorization into two binomials.
This approach is especially useful for high-degree polynomials and is often used before applying AC or other formulas.
4) Worked Examples from the Lecture
Example A: 81 c^4 + 90 c^2 d + 25 d^2
Observation: This is a perfect square trinomial of the form A^2 + 2AB + B^2 with A = 9c^2 and B = 5d.
Factorization:81 c^4 + 90 c^2 d + 25 d^2 = (9c^2 + 5d)^2.
The middle term $90 c^2 d$ corresponds to $2AB = 2(9c^2)(5d) = 90 c^2 d.$
Example B: $x^3 - 27$
Recognize as a difference of cubes with a = x and b = 3 since 27 = 3^3.
Factorization:x^3 - 27 = (x-3)(x^2 + 3x + 9).
Example C: $8m^6 + 27n^3$
Recognize as a sum of cubes with a = 2m^2 and b = 3n since (2m^2)^3 = 8m^6 and (3n)^3 = 27n^3.
Factorization: 8m^6 + 27n^3 = (2m^2 + 3n)(4m^4 - 6m^2n + 9n^2).
Example D: Difference of squares with a two-variable twist
Example: p^2 - 100 = p^2 - 10^2 = (p-10)(p+10).
If a polynomial factors as a difference of squares in the form A^2 - B^2, apply the same rule: (A-B)(A+B).
Example E: Difference of squares in a single-variable quadratic
Example: x^2 - 49 = (x-7)(x+7).
Quick note on sums of cubes that do not appear in the lecture text
For a sum of cubes, remember: a^3 + b^3 = (a+b)(a^2 - ab + b^2).
For a difference of cubes: a^3 - b^3 = (a-b)(a^2 + ab + b^2).
Relationship to higher-degree polynomials
The lecture emphasizes recognizing patterns that fit these standard formulas so you can factor quickly and correctly, especially before moving into more advanced topics like fractions and complex fractions.
5) Rational Expressions: Definitions, Restrictions, and Simplification
What is a rational expression?
A ratio of two polynomials: \frac{P(x)}{Q(x)}, with Q(x) \neq 0. In multivariable cases, denominators involve expressions in the variables and are not allowed to be zero.
Restrictions (domain considerations)
You must exclude any x-values that make the denominator zero.
For multiple variables, you exclude combinations that make any denominator zero (e.g., if a fraction has factors like $(x)(y)$ in the denominator, you require x \neq 0 and y \neq 0).
Basic simplifying approach
Factor the numerator and the denominator.
Cancel common factors that appear in both numerator and denominator.
Keep track of domain restrictions: the original expression can restrict certain values even if they cancel in the simplified form.
Multiplication and division of rational expressions
Multiplication: multiply numerators together and multiply denominators together, then simplify.
Division: multiply by the reciprocal of the divisor (i.e., convert division to multiplication by the reciprocal) and then simplify. Restrictions include that all denominators involved must be nonzero.
In variables, you must maintain the restrictions on all denominators throughout the operation.
Complex fractions
A complex fraction is a fraction whose numerator or denominator also contains fractions.
Strategy: simplify by turning the complex fraction into a product by multiplying by the reciprocal, then simplify by factoring and canceling common factors.
Practical example (from the lecture)
Example: Simplify \frac{x^2-8x}{x^2-7x-8}.
Factor: x^2-8x = x(x-8),
x^2-7x-8 = (x-8)(x+1).Cancel the common factor (x-8) to obtain \frac{x}{x+1}.
Domain restrictions: exclude values that make the original denominator zero, i.e., x \neq 8, -1.
Additional practical notes
When simplifying expressions with variables, always state the restricted values of the denominator before and after cancellation.
In more advanced contexts you may encounter vertical/horizontal asymptotes in graphs, but those are covered in later coursework.
6) Practice and Problem-Solving Strategies
Factoring by inspection vs. systematic methods
When coefficients are simple, you can often factor by inspection (trial and error for the middle term, or recognizing a pattern).
For quadratics with a leading coefficient not equal to 1, use the AC method to guide splitting the middle term.
Factoring by grouping
Useful for four-term polynomials: group terms to extract a common factor in each group, then factor the resulting binomial common factor.
Recognizing patterns early
The fastest problems are those where the expression clearly matches a known pattern (difference of squares, difference of cubes, sum of cubes, or a perfect square trinomial).
Verification by expansion
After proposing a factorization, expand to verify you reproduce the original polynomial.
Domain awareness when simplifying
Always keep track of restrictions on the domain; a canceled factor may hide an excluded value in the original expression.
7) Class Logistics and Exam Preparation (Contextual Notes)
Homework and practice
Homework is open until Sunday; practice problems align with the material covered so far.
Expect a practice test with a time limit; the test may be delivered under lockdown browser conditions.
Section progression
The plan includes moving to Hour 6 and preparing for a test; the material covers factoring strategies, rational expressions, and simplification.
Test strategy overview
-Problems will likely be narrowed for practice; focus on quick recognition of factoring patterns and safe factoring by inspection.Key takeaways for exams
Be able to identify: difference of squares, difference of cubes, sum of cubes, perfect-square trinomials, and apply AC method for quadratics.
Be comfortable with simplifying rational expressions and performing operations with fractions, including complex fractions.
Always determine restrictions before simplifying or solving.
8) Quick Reference Formulas to Memorize
Difference of squares: a^2 - b^2 = (a-b)(a+b)
Square of a binomial: (A+B)^2 = A^2 + 2AB + B^2
Difference of cubes: a^3 - b^3 = (a-b)(a^2 + ab + b^2)
Sum of cubes: a^3 + b^3 = (a+b)(a^2 - ab + b^2)
AC method: find $m,n$ with m+n=b, ext{ and } mn=ac; rewrite as ax^2+mx+nx+c and factor by grouping.
Rational expression basics:
$\dfrac{P(x)}{Q(x)}$ with $Q(x)\neq0$; determine domain by setting $Q(x)=0$ and solving for excluded values.
For multiplication/division, cancel common factors after factoring; for division, multiply by the reciprocal.
Notation used in examples
Perfect-square binomial with $A=9c^2$, $B=5d$ leads to $(9c^2+5d)^2$.
Difference of cubes with $a=x$, $b=3$ leads to $(x-3)(x^2+3x+9)$.
Sum of cubes with $a=2m^2$, $b=3n$ leads to $(2m^2+3n)(4m^4-6m^2n+9n^2)$.
If you want, I can tailor a condensed cheat-sheet focusing on the exact problem types you expect on the exam (e.g., a quick reference for AC method steps and a few worked-by-pattern examples). Let me know the topics you want emphasized or any particular problem set you’d like included in the notes.