Polynomial Operations: FOIL, Binomial/Trinomial Products, Division by Monomial, and Geometric Applications

  • Multiplying Binomials (FOIL)

    • 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 the FOIL method: First, Outer, Inner, Last.

    • FOIL formula: (a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd

    • Example 1: (x+2)(x+4)=x2+6x+8(x + 2)(x + 4) = x^2 + 6x + 8

    • Example 2: (2x+1)(x−5)=2x2−9x−5(2x + 1)(x - 5) = 2x^2 - 9x - 5

    • Practice: You Try! Problems (Final answers must be in standard form)

    1. (y + 8)(y + 1)

    2. (k - 4)(k + 5)

    3. (x - 10)(x - 4)

    4. (x + 2)(x - 2)

    5. (4x - 7)(x + 3)

    6. (n - 1)(5n - 4)

    7. (3y + 1)(3y + 2)

    8. (6a + 2)(2a + 3)

    9. (8h - 3)(3h - 1)

    10. (3w + 1)(3w - 1)

    • Then more practice (same idea): problems 11–18 include binomial × binomial, binomial × binomial with negative signs, and binomial squares.

    • Binomial × Binomial / Special products (11–18):

    1. ((5x + 4y)(x - y))

    2. ((2a + 5b)(a - 4b))

    3. ((2r + s)(2r - s))

    4. ((4c + d)(7c - 2d))

    5. ((x + 4)^2)

    6. ((p - 7)^2)

    7. ((2m - 5)^2)

    8. ((a + 3b)^2)

    • Binomial × Trinomial (19–22):

    1. ((x + 4)(x^2 + 3x - 6))

    2. ((k - 5)(k^2 - k - 8))

    3. ((3a + 1)(5a^2 + 2a - 6))

    4. ((2v + 3)(4v^2 - 3v - 6))

    • Geometric Application (23): Find the area of the shaded region as a simplified expression. The grid shows four algebraic expressions: (9x - 2), (x + 12), (3x), and (2x + 7). A common setup is a difference of two rectangle areas:
      A=(9x−2)(2x+7)−(x+12)(3x).A = (9x - 2)(2x + 7) - (x + 12)(3x).
      Simplified: A=15x2+23x−14.A = 15x^2 + 23x - 14.

  • Dividing a Polynomial by a Monomial

    • Key idea: To divide a polynomial by a monomial, divide each term in the polynomial by the monomial, applying the quotient rule to any x-powers.

    • Quotient Rule (for single-variable x):axmbxn=abxm−n\frac{a x^m}{b x^n} = \frac{a}{b} x^{m-n}

    • Example 1: 28x44x=7x3\frac{28x^4}{4x} = 7x^3

    • Example 2: (illustrative) 12x2y2xy2=6xy=6xy\frac{12x^2 y}{2xy^2} = 6 \frac{x}{y} = \frac{6x}{y}

    • You Try! (Divide each term by the monomial using the quotient rule; practice problems are provided in the transcript.)

  • Further practice: Challenging problems (Put It All Together)

    • Directions: Simplify each expression completely.

    • These problems combine multiplication of polynomials with distribution and potentially subtraction of areas or combining like terms.

    • Problems 17–20 are multi-step and require careful FOIL/distribution and combining like terms across multiple factors.

    • Examples of the types of work involved include distributing products across binomials and trinomial factors, then simplifying to a standard polynomial form.

  • Quick reference: core formulas you’ll use

    • FOIL: (a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd

    • Special products:

    • Square of a binomial: (x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2

    • Difference of squares: (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2

    • Binomial × Trinomial: distribute each binomial term across the trinomial and combine like terms.

  • Tips for success on the exam

    • Always write in standard form, arranging terms by decreasing powers of the main variable.

    • Check your sign arithmetic carefully, especially in problems with minus signs.

    • For geometric-area-type problems, consider setting up as a difference of two rectangular areas when there is a grid of lengths; then apply FOIL to each product before simplifying.

    • When dividing by a monomial, split the division term-by-term and apply the exponent rules consistently.