6.3 Multiply Polynomials

Multiplying polynomials is a key skill in algebra. It's all about combining terms in specific ways to create more complex expressions. This process builds on basic arithmetic, applying those concepts to variables and exponents.

Understanding polynomial multiplication opens doors to solving real-world problems. From calculating areas to modeling complex systems, these techniques are essential. They form the foundation for more advanced math concepts you'll encounter later.

Multiplying Polynomials

Polynomial-monomial multiplication

• Multiply each term of the polynomial by the monomial
  • Multiply coefficients of monomial and each polynomial term ($3x \cdot 4x^2 = 12x^3$)
  • Add exponents of like bases (variables) in monomial and each polynomial term ($x^2 \cdot x = x^3$)
  • Combine like terms after multiplying, if applicable ($3x^2 + 2x^2 = 5x^2$)
• Example: Multiply $2x^3 - 3x + 4$ by $-5x$
  1. $-5x(2x^3) = -10x^4$
  2. $-5x(-3x) = 15x^2$
  3. $-5x(4) = -20x$
  • Result: $-10x^4 + 15x^2 - 20x$

FOIL method for binomials

• FOIL: First, Outer, Inner, Last - order of multiplying terms in two binomials (also known as algebraic expansion)
  • First: Multiply first terms of each binomial ($x \cdot x = x^2$)
  • Outer: Multiply outer terms - first term of first binomial and second term of second binomial ($x \cdot 4 = 4x$)
  • Inner: Multiply inner terms - second term of first binomial and first term of second binomial ($3 \cdot x = 3x$)
  • Last: Multiply last terms of each binomial ($3 \cdot 4 = 12$)
• Add resulting terms, combining like terms if necessary ($4x + 3x = 7x$)
• Example: Multiply $(x - 5)(x + 2)$ using FOIL
  1. First: $x \cdot x = x^2$
  2. Outer: $x \cdot 2 = 2x$
  3. Inner: $-5 \cdot x = -5x$
  4. Last: $-5 \cdot 2 = -10$
  • Add terms: $x^2 + 2x - 5x - 10$
  • Combine like terms: $x^2 - 3x - 10$

Trinomial-binomial expansion

• Use distributive property to multiply trinomial by binomial
  • Multiply each term of trinomial by each term of binomial ($x^2(x) = x^3$, $x^2(3) = 3x^2$)
  • Add resulting terms, combining like terms if necessary ($2x^2 + x^2 = 3x^2$)
• Example: Multiply $(2x^2 - 3x + 1)(x - 4)$
  1. Multiply $2x^2$ by $(x - 4)$: $2x^2(x) = 2x^3$ and $2x^2(-4) = -8x^2$
  2. Multiply $-3x$ by $(x - 4)$: $-3x(x) = -3x^2$ and $-3x(-4) = 12x$
  3. Multiply $1$ by $(x - 4)$: $1(x) = x$ and $1(-4) = -4$
  • Add resulting terms: $2x^3 - 8x^2 - 3x^2 + 12x + x - 4$
  • Combine like terms: $2x^3 - 11x^2 + 13x - 4$

Alternative Multiplication Methods

• Vertical multiplication: Arrange polynomials vertically and multiply term by term, similar to long multiplication with numbers
• Area model: Use a rectangular grid to represent the product of polynomials, with each cell representing the product of two terms
• Pascal's triangle: Useful for expanding binomials raised to powers, showing coefficients of expanded terms
• Factoring: The reverse process of polynomial multiplication, used to simplify or solve polynomial equations