Multiplying polynomials is a key skill in algebra. It involves combining simpler expressions to create more complex ones. From multiplying monomials to tackling binomials and beyond, each step builds on the last.

The process uses rules like FOIL and distribution to expand expressions. Special product forms, like perfect squares and differences of squares, offer shortcuts. Understanding polynomial structure helps in both multiplication and factoring.

Multiplying Polynomials

Multiplication of monomials

• Multiply coefficients of monomials
• Add exponents of like variables using product rule: $x^a \cdot x^b = x^{a+b}$ ($3x^2 \cdot 4x^3 = 12x^5$)
• Simplify resulting monomial by combining like terms

Polynomial by monomial multiplication

• Distribute monomial to each term of polynomial
  • Multiply coefficient of each polynomial term by monomial coefficient
  • Add exponents of like variables using product rule
• Simplify resulting polynomial by combining like terms ($3x^2(2x^3 + 4x - 1) = 6x^5 + 12x^3 - 3x^2$)

Binomial multiplication process

• Use FOIL method (First, Outer, Inner, Last) to multiply binomials
  1. Multiply first terms
  2. Multiply outer terms
  3. Multiply inner terms
  4. Multiply last terms
  • $(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6$
• Simplify resulting polynomial by combining like terms

Multiplication of varied polynomials

• Distribute each term of first polynomial to each term of second polynomial
  • Multiply coefficients and add exponents of like variables for each pair of terms
• Simplify resulting polynomial by combining like terms
  • $(2x^2 + 3x - 1)(x + 2) = 2x^3 + 4x^2 + 3x^2 + 6x - x - 2 = 2x^3 + 7x^2 + 5x - 2$
• The result is in expanded form, showing all terms explicitly

Special product forms

• Perfect square binomials: $(a \pm b)^2 = a^2 \pm 2ab + b^2$ ($(x + 3)^2 = x^2 + 6x + 9$)
• Difference of squares: $(a^2 - b^2) = (a + b)(a - b)$ ($(x^2 - 9) = (x + 3)(x - 3)$)
• Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Products of polynomial functions

• Substitute given value for variable in each polynomial function
• Multiply resulting polynomials using appropriate methods (monomial, binomial, or general polynomial multiplication)
• Simplify final result by performing arithmetic operations
  • If $f(x) = 2x + 1$ and $g(x) = x - 3$, find $f(2) \cdot g(2)$
    1. $f(2) = 2(2) + 1 = 5$ and $g(2) = 2 - 3 = -1$
    2. $f(2) \cdot g(2) = 5 \cdot (-1) = -5$

Understanding polynomial structure

• The degree of a polynomial is the highest power of the variable in the polynomial
• The constant term is the term without a variable (e.g., the -2 in $2x^3 + 7x^2 + 5x - 2$)
• Factoring is the reverse process of polynomial multiplication, breaking down a polynomial into its factors