Math Day 3

Multiplying Polynomials

• Core idea: use the distributive law to combine polynomials.
• FOIL (for binomials) = First, Outer, Inner, Last to multiply two binomials.
• Example 1: 3x times a trinomial
  • Compute: $3x\cdot(x^2+5x+1)=3x\cdot x^2 + 3x\cdot 5x + 3x\cdot 1 = 3x^3 + 15x^2 + 3x.$
• Example 2: (x+1)(x-2)
  • FOIL: $x\cdot x + x\cdot (-2) + 1\cdot x + 1\cdot (-2) = x^2 - 2x + x - 2 = x^2 - x - 2.$
• Note on transcript errors: Distributive law is correct; explicit expansions should follow standard algebra rules (e.g., 3x(x^2+5x+1) = 3x^3 + 15x^2 + 3x).

Distributive Law, Factoring, and Basic Identities

• Distributive law in polynomials: for any polynomials A, B, C, we have A(B+C)=AB+AC.
• When multiplying, distribute each term of one factor across the other.
• Opposite of multiplying: factoring - breaking a polynomial into a product of simpler polynomials.
• Example of cube of a binomial (to be used later):
  • For a=2x, b=y, the cube (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
  • So, $(2x - y)^3 = 8x^3 - 12x^2y + 6xy^2 - y^3.$

Factoring Basics

• Look for common factors first:
  • Example: $2n^3 + 4n^2 + 6n = 2n(n^2 + 2n + 3).$
• Difference of squares:
  • Identity: $a^2 - b^2 = (a+b)(a-b).$
  • Example: $25x^2 - y^2 = (5x)^2 - (y)^2 = (5x + y)(5x - y).$
• Recognize patterns to simplify factoring before attempting more work.

Factoring Trinomials (Monic Case)

• For a trinomial of the form $x^2 + bx + c$, we seek two numbers that multiply to c and add to b.
• Example: Factor $x^2 - 2x - 15$
  • We need two numbers whose product is $c=-15$ and whose sum is $b=-2$.
  • Pairs to consider: \{-1,15 o 14; 1,-15 o -14; -3,5 o 2; 3,-5 o -2}.
  • The pair that sums to -2 is $-5\ and\ 3.$
  • Therefore, $x^2 - 2x - 15 = (x - 5)(x + 3).$

Why this works

• FOIL to check: $(x - 5)(x + 3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15.$
• The constant term is c, and the product of the two numbers equals c; their sum equals b.

Special Case: Perfect Square Trinomials

• Pattern: $a^2 \,\pm\, 2ab + b^2$ factor as $(a \pm b)^2$
• Example: $x^2 - 2x + 1 = (x - 1)^2.$
• Reason: $(x - 1)^2 = x^2 - 2x + 1.$

Factoring with two-variable quadratics (in x and y)

• When factoring expressions like $ax^2 + bxy + cy^2$, look for two binomials: $(px + qy)(rx + sy)$
• Example: Factor $6x^2 - 25xy + 4y^2$
  • View as a quadratic in x: $6x^2 + (-25y)x + 4y^2.$ Then ac = 6*4y^2 = 24y^2 and need two numbers that multiply to 24y^2 and sum to -25y.
  • Choose two terms -24y and -1y, since (-24y) + (-1y) = -25y and (-24y)(-1y) = 24y^2.
  • Rewrite: $6x^2 - 24xy - xy + 4y^2.$
  • Group: $6x(x - 4y) - y(x - 4y) = (6x - y)(x - 4y).$
  • Therefore, $6x^2 - 25xy + 4y^2 = (6x - y)(x - 4y).$

Factoring by a Common Factor and by Grouping (specific example)

• From transcript examples:
  • Factor out the greatest common factor when present.
  • Use factoring by grouping when c is not easily factored by simple GCF or simple trinomial methods.

Factoring by Recognizing Special Products in 2-variables

• Two-variable square-trinomial patterns and cross-term strategies help factor as products of binomials like (Ax + By)(Cx + Dy).

Sum and Difference of Cubes

• Core identities:
  • Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2).$
  • Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2).$
• Examples:
  • Difference of cubes: $8x^3 - 1 = (2x)^3 - 1^3 = (2x - 1)(4x^2 + 2x + 1).$
  • Sum of cubes: $8x^3 + 1 = (2x + 1)(4x^2 - 2x + 1).$
• Important note: In transcript, there were misstatements about exponents (e.g., 8x^2 rather than 8x^3). The correct forms above follow from the cube identities.

Quick Reference: Common Factoring Routes

• If there is a common factor: factor it out first.
• If there is a quadratic in x (or y) with leading coefficient 1: look for two numbers that multiply to c and add to b.
• If a quadratic has two variables: try to factor as a product of two binomials; use ac-method (split ac into two terms that sum to b).
• For a^2 - b^2: use difference of squares identity.
• For a^3 ± b^3: use sum/difference of cubes identities.

Real-World Relevance and Connections

• Polynomial multiplication is foundational for algebra, calculus (polynomial integrals/derivatives), and solving polynomial equations.
• Factoring helps solve polynomial equations by turning them into products equal to zero.
• Understanding special products (difference of squares, perfect square trinomials, sum/difference of cubes) speeds up factoring and supports pattern recognition in higher-level math.
• These techniques connect to polynomial identities used in transforming and simplifying expressions in physics, engineering, and computer science (e.g., simplifying signals, factoring characteristic polynomials).

Worked Examples to Practice

• Multiply: $3x(x^2 + 5x + 1)$
  • Result: $3x^3 + 15x^2 + 3x$
• Factor: $x^2 - 2x - 15$
  • Two numbers: -5 and 3; factors: $(x - 5)(x + 3)$
• Factor: $6x^2 - 25xy + 4y^2$
  • Factorization: $(6x - y)(x - 4y)$
• Factor: $8x^3 - 1$
  • Difference of cubes: $(2x - 1)(4x^2 + 2x + 1)$
• Factor: $8x^3 + 1$
  • Sum of cubes: $(2x + 1)(4x^2 - 2x + 1)$
• Factor: $x^2 - 2x + 1$
  • Perfect square: $(x - 1)^2$
• Expand: $(x + 1)(x - 2)$ to verify: $x^2 - x - 2$

Note: All numerical results and identities above are corrected for consistency with standard algebraic rules. The transcript contained several typographical errors; the notes above present the correct forms and typical solving strategies.