All Things Algebra: Polynomial Multiplication & Division Notes (FOIL, special products, and monomial division)

Multiplying Binomials

  • Main idea: To multiply binomials, distribute each term in the first binomial to each term in the second binomial, then combine like terms. This is commonly called FOIL: First, Outer, Inner, Last.
  • FOIL formula:
    (a+b)(c+d)=ac+ad+bc+bd
  • Examples
    • Example 1: (x+2)(x+4)=x^2+6x+8
    • Example 2: (2x+1)(x-5)=2x^2-9x-5

You Try! (Discrete problems from the transcript)

  • 1. (y+8)(y+1)=y^2+9y+8

  • 2. (k-4)(k+5)=k^2+k-20

  • 3. (x-10)(x-4)=x^2-14x+40

  • 4. (x+2)(x-2)=x^2-4

  • 5. (4x-7)(x+3)=4x^2+5x-21

  • 6. (n-1)(5n-4)=5n^2-9n+4

  • 7. (3y+1)(3y+2)=9y^2+9y+2

  • 8. (6a+2)(2a+3)=12a^2+22a+6

  • 9. (8h-3)(3h-1)=24h^2-17h+3

    1. (3w+1)(3w-1)=9w^2-1

  • Name: Class: Date: Topic:

Binomial × Binomial / Binomial × Trinomial

  • Directions: Find each product. Final answers must be in standard form.

    1. $(5x+4y)(x-y)=5x^2-xy-4y^2$
    1. $(2a+5b)(a-4b)=2a^2-3ab-20b^2$
    1. $(2r+s)(2r-s)=4r^2-s^2$
    1. $(4c+d)(7c-2d)=28c^2-cd-2d^2$
    1. $(x+4)^2=x^2+8x+16$
    1. $(p-7)^2=p^2-14p+49$
    1. $(2m-5)^2=4m^2-20m+25$
    1. $(a+3b)^2=a^2+6ab+9b^2$
    1. $(x+4)(x^2+3x-6)=x^3+7x^2+6x-24$
    1. $(k-5)(k^2-k-8)=k^3-6k^2-3k+40$
    1. $(3a+1)(5a^2+2a-6)=15a^3+11a^2-16a-6$
    1. $(2v+3)(4v^2-3v-6)=8v^3+6v^2-21v-18$

  • Geometric Application (Area of shaded region)

    • Given diagram (dimensions inferred from transcript): large rectangle with dimensions $(9x-2)$ by $(x+12)$ minus a smaller rectangle with dimensions $3x$ by $(2x+7)$.
    • Plan: area of shaded region = area of large rectangle − area of small rectangle.
    • Calculation:
      A1=(9x-2)(x+12)=9x^2+106x-24 A2=3x(2x+7)=6x^2+21x
      A{ ext{shaded}}=A1-A_2=(9x^2+106x-24)-(6x^2+21x)=3x^2+85x-24
    • Final expression: A_{ ext{shaded}}=3x^2+85x-24

Dividing a Polynomial by a Monomial

  • Core idea: To divide a polynomial by a monomial, divide each term of the numerator by the monomial in the denominator, applying the quotient rule to variables for exponents.

  • Quotient Rule (for variables):
    rac{x^a}{x^b}=x^{a-b}
    ext{ (for }x
    eq0 ext{)}

  • Examples (illustrative, consistent with the rule):

    • rac{28x^4}{4x}=7x^3
    • rac{12x^3y^2}{6xy}=2x^2y

  • Note: The transcript contains several garbled examples; the essential rule remains dividing each term and applying exponent subtraction per the quotient rule.

  • Practice structure in the transcript: problems 1–10 are quotients to compute; problems 11–20 (in later pages) extend to more quotients and combined simplifications. The main objective is to become fluent with term-by-term division and exponent arithmetic.

  • Quick tips:

    • Break apart numerator terms: for each term, write term ÷ monomial, then combine results.
    • Keep track of coefficients and variables separately until the end.
    • When subtracting exponents, ensure you only apply to like bases; if a base is missing in a term, it contributes a factor of 1.

  • Challenge/Put It All Together (summary approach):

    • Step 1: Identify all polynomial factors and the monomial divisor.
    • Step 2: Divide each term in the numerator by the divisor, applying the quotient rule to exponents.
    • Step 3: Collect all quotients into a single simplified expression.
    • Step 4: If there are like terms, combine them; check for further simplification.

  • Ethical/Philosophical/Practical implications: This section is a pure algebraic manipulation exercise; the practical relevance lies in foundational skills for higher-level math, physics, engineering, and data analysis. The content emphasizes method, accuracy, and step-by-step reasoning rather than interpretation or opinion.

Put It All Together (Challenge) — Notes on approach

  • The Challenge problems require integrating multiple previous skills: FOIL/distributive expansion, combining like terms, and sometimes factoring or recognizing patterns (e.g., difference of squares).

  • General strategy:

    • Expand every product fully using distributive property or FOIL.
    • Collect like terms across the expanded expression.
    • Factor or simplify if the instruction asks for a reduced form or a factorization.

  • Example workflow (illustrative, not from transcript):

    • Multiply two binomials: $(2x+3)(x+4)$ → expand to $2x^2+8x+3x+12=2x^2+11x+12$.
    • If there is subtraction of expressions, perform the subtraction after full expansion.

  • Final reminder: The materials in the transcript include a lot of concrete problems (Problems 1–22) and a geometric application (Problem 23). Where exact phrasing of some later problems is garbled in the transcript, the essential methods are unchanged and are captured above for practice and study.