Factoring Polynomials and Rational Expressions — Study Notes
Homework reminders
Friday: Homework 1 is due tonight. It’s designed to be short and to help you learn how you’ll submit assignments for the rest of the semester.
Homework 2 will be posted this weekend (today or tomorrow) and will be due next Friday.
The goal of these assignments is to practice the submission workflow for the course, not just to test you on problem solving.
Recap: Factoring polynomials (one- and two-variable)
Goal: Factor polynomials; start with the greatest common factor (GCF) when present to simplify.
General strategy when factoring a polynomial in multiple variables:
Identify the GCF among all terms (including variables, taking the lowest exponent of each variable that appears).
Factor out the GCF.
Factor the remaining polynomial (often a quadratic, but can be higher degree) by standard methods (factoring by grouping, special forms, etc.).
Example 1: Factor a two-variable quadratic
Given: 12x^{2}y - 10xy - 12y
Step 1: Factor out the GCF from all terms.
Coefficients: gcd(12, 10, 12) = 2; variables: both contain y, x appears in the first two terms but not the last.
GCF = 2y
Factor out GCF: 2y(6x^{2} - 5x - 6)
Now factor the inner quadratic by grouping:
For 6x^{2} - 5x - 6, compute a c = 6 imes (-6) = -36 and find two numbers that multiply to -36 and sum to -5: -9 and 4.
Rewriting: 6x^{2} - 9x + 4x - 6
Factor by grouping: 3x(2x - 3) + 2(2x - 3) = (2x - 3)(3x + 2)
Final factorization: 2y(2x - 3)(3x + 2)
Special forms to recognize:
Difference of two cubes: a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})
Difference of two squares: a^{2} - b^{2} = (a + b)(a - b)
Sum of two squares cannot be factored over the integers: a^{2} + b^{2} is irreducible in this context (prime).
Example note: If you see something like x^{2} + 1 or x^{2} + 5x + 1 with no obvious factorization, check if it’s a prime quadratic in simplest form.
Example 2: Factoring a product that becomes a difference of squares
Given: u^{2}u^{2}v - (something) (transcript excerpt shows factoring technique)
Conceptual steps demonstrated: factor GCF first, rewrite as a difference of squares if possible, then apply the difference-of-squares formula. In the discussed example the GCF was u^{2}v, and the expression was rewritten as a difference of squares: u^{2}vig((2u)^{2} - (3v)^{2}ig) = u^{2}v(2u - 3v)(2u + 3v).
Additional notes from the lesson:
When you have a product with multiple terms, always check for a common factor across all terms first.
If you can rewrite a part of the polynomial as a difference of squares, apply the identity to factor further.
When factoring a sum of two cubes or a difference of two cubes, apply the respective formulas and then look for common factors to extract.
Rational expressions: definitions and reduction (Section 1.4 overview)
What is a rational expression?
A ratio of two polynomials where both numerator and denominator are polynomials (the denominator is not zero).
Can involve polynomials in one or more variables.
Goal: simplify (reduce to lowest terms) by canceling common factors after fully factoring the numerator and denominator.
Recommended procedure to reduce rational expressions:
Factor the numerator completely.
Factor the denominator completely.
Cancel any common factors that appear in both numerator and denominator.
Check the result to ensure no further reduction is possible.
Important caution: You can only cancel factors, not terms; cancellation must reflect equal factors in numerator and denominator, not just common numerical factors unless they are factors of the polynomials involved.
Examples: simplifying rational expressions
Example A (cancellation with GCF):
Expression: rac{3x(x - 5)}{6x^{2}}
Factor the numerator: already factored as 3x(x - 5)
Factor the denominator: 6x^{2} = 3 imes 2 imes x^{2} = 3x imes 2x
Cancel common factor 3x:
Result: rac{x - 5}{2x}
Example B (cancel a common binomial factor):
Expression: rac{(x + 1)(x + 2)}{x + 1}
Cancel the common factor x + 1
Result: x + 2
Example C (more complex top and bottom): (from transcript)
Top: x^{2} + 4xy - 5y^{2}
Denominator: claim of factoring into a product including (x - y) factors. A clean way to mirror the example in the notes:
Factor the numerator: x^{2} + 4xy - 5y^{2} = (x + 5y)(x - y)
Suppose the denominator factors as 6x(x + y)(x - y) (as discussed in the lecture). Then cancel the common factor x - y to obtain:
Result: rac{x + 5y}{6x(x + y)}
(Note: this setup is chosen to illustrate the cancellation of a common binomial factor; always verify the actual denominator factors in your specific problem.)
Multiplying and dividing rational expressions
Multiplication rule: (P/Q) * (R/S) = (PR) / (QS).
Division rule: (P/Q) ÷ (R/S) = (P/Q) * (S/R) = (PS) / (QR).
Practical approach:
Always factor all rational expressions first.
Cancel any common factors before performing the multiplication or division to keep the fractions reduced.
Example A (multiplication with a common factor):
Given: (4x^2 - 25) * 14 / (2x + 5)
Factor 4x^2 - 25 as a difference of squares: (2x + 5)(2x - 5)
Cancel the common factor (2x + 5):
Result before final simplification: 14(2x - 5) = 28x - 70
Final reduced result: 28x - 70
Example B (division with cubes and squares):
A more involved example from the lecture used factoring of sums/differences of cubes and squares, followed by cancellation after rewriting with a common denominator. The key idea is to cancel identical factors across numerator and denominator before performing the multiplication by the reciprocal (for division).
Practical tip: Always perform factoring and cancellation before performing the arithmetic, and aim for the reduced form at the end.
Addition and subtraction of rational expressions
When denominators are the same:
Add or subtract numerators while keeping the common denominator.
Example: If you have rac{A}{D} + rac{B}{D} = rac{A + B}{D} and similarly for subtraction: rac{A}{D} - rac{B}{D} = rac{A - B}{D}.
After combining, simplify if possible by factoring the numerator and canceling any common factors with the denominator.
When denominators are different: find a common denominator, specifically the least common denominator (LCD).
Steps to find the LCD:
Factor each denominator completely.
List all distinct factors appearing in any denominator, each raised to the highest power it appears with in any denominator.
Multiply these factors to obtain the LCD.
Then rewrite each fraction with the LCD as its denominator and combine.
Example: Adding/subtracting with LCD
Given fractions with denominators 8x and 4x^2:
Denominators: 8x = 2^3 x, 4x^2 = 2^2 x^2
LCD is 8x^2 (max powers: 2^3 and x^2)
Rewrite each expression with denominator 8x^2:
First fraction:
If it is x - 4 over 8x, multiply numerator and denominator by x to get (x^2 - 4x) / (8x^2)
Second fraction:
If it is (x - 2) over 4x^2, multiply numerator and denominator by 2 to get (2x - 4) / (8x^2)
Combine: (x^2 - 4x - (2x - 4)) / (8x^2) = (x^2 - 6x + 4) / (8x^2)
Another LCD example (more complex):
Example denominators: 6x^2, 3x^5, 2x^7
LCD calculation: factor each denominator and take the highest powers: 6x^2 = 2·3·x^2, 3x^5 = 3·x^5, 2x^7 = 2·x^7 → LCD = 6x^7
Rewrite each fraction with denominator 6x^7 and perform addition/subtraction:
5/(6x^2) → 5x^5/(6x^7)
2/(3x^5) → 4x^2/(6x^7)
7/(2x^7) → 21/(6x^7)
Combined result: rac{5x^5 + 4x^2 + 21}{6x^7}
Check for possible simplification: gcd of coefficients {5, 4, 21} is 1, so no common factor to cancel with the denominator’s factors of 6 and x; the fraction is in lowest terms.
Real-world and exam-oriented notes
Practice tips:
Always start with the GCF when factoring; it simplifies later steps.
After factoring, consider special forms (difference of squares, sum of cubes, etc.) to complete the factorization.
For rational expressions, always factor everything first, then cancel common factors before performing any arithmetic.
When adding or subtracting with unlike denominators, find the LCD and rewrite each fraction to that common denominator before combining.
Common mistakes to avoid:
Forgetting to cancel only common factors, not whole terms.
Canceling terms that are not exact factors (e.g., canceling x with x+1).
Missing the highest powers when computing the LCD; ensure you pick the maximum exponent of each factor across all denominators.
Practical implications:
These techniques reduce complex algebraic expressions, which in turn simplifies solving equations, simplifying integrands in calculus, and manipulating expressions in physics/engineering problems.
Quick recap on the flow of solving problems in this unit
Start with identifying and factoring out the GCF.
Use grouping or special formulas to factor quadratics like ax^2 + bx + c by looking for two numbers p and q such that pq = ac and p + q = b.
Recognize special patterns: difference of squares, difference/sum of cubes.
In rational expressions, factor numerator and denominator completely, cancel common factors, then perform the required operation (multiply/divide/add/subtract).
When adding or subtracting with different denominators, compute the LCD, rewrite each expression with the LCD, then combine and simplify.
Note: The transcript contained a few arithmetic slips in a worked example (e.g., a result that should be 28x − 70 rather than 28x − 7). The methods and final forms shown in these notes follow standard algebra rules and are intended to reflect correct factoring and simplification steps. If you see a mismatch in a specific worked example on the test, use the factoring steps outlined above to verify the correct result.