Polynomials (add, multiply, factor) Knowledge-Based Study

Study notes

1. What is a Polynomial?

   A polynomial is an algebraic expression consisting of variables and coefficients, combined using only addition, subtraction, and non-negative integer exponents of variables.

   A polynomial in one variable, $x$, is commonly written in standard form with descending powers of $x$:

   $$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$

   Adding and Subtracting Polynomials

   To add or subtract polynomials, identify and combine like terms. Like terms are terms that share identical variable parts raised to the exact same exponents.

   • Addition: Remove grouping symbols and combine coefficients of like terms.
   • Subtraction: Distribute the negative sign across every term of the polynomial being subtracted, then combine like terms.

   Example 1: Simplify $(3x^2 - 5x + 7) + (2x^2 + 8x - 4)$.

   Group like terms:

   $$(3x^2 + 2x^2) + (-5x + 8x) + (7 - 4) = 5x^2 + 3x + 3$$

   Example 2: Simplify $(6x^3 + 4x - 9) - (2x^3 - 3x^2 + 7)$.

   Distribute the negative sign and combine:

   $$6x^3 + 4x - 9 - 2x^3 + 3x^2 - 7 = 4x^3 + 3x^2 + 4x - 16$$

   The Distributive Property in Polynomial Multiplication

   Multiplying polynomials relies on the distributive property, $a(b + c) = ab + ac$, together with the product rule of exponents: $x^m \cdot x^n = x^{m+n}$.

   Multiplying a Monomial by a Polynomial

   Multiply each term in the polynomial by the monomial factor:

   $$-2x^2(4x^3 - 5x + 1) = (-2x^2 \cdot 4x^3) + (-2x^2 \cdot -5x) + (-2x^2 \cdot 1) = -8x^5 + 10x^3 - 2x^2$$

   Multiplying Binomials

   To multiply two binomials $(a + b)(c + d)$, distribute every term in the first factor to every term in the second factor:

   $$(a + b)(c + d) = ac + ad + bc + bd$$

   Example: Multiply $(2x + 3)(x - 5)$.

   $$(2x)(x) + (2x)(-5) + (3)(x) + (3)(-5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15$$

   Multiplying Larger Polynomials

   Multiply each term of the first polynomial by each term of the second polynomial systematically, then combine like terms.

   Example: Expand $(x - 2)(3x^2 + 4x - 1)$.

   $$x(3x^2 + 4x - 1) - 2(3x^2 + 4x - 1) = (3x^3 + 4x^2 - x) - (6x^2 + 8x - 2)$$

   $$= 3x^3 - 2x^2 - 9x + 2$$

   Factoring Polynomials

   Factoring is the reverse process of polynomial multiplication. A factored polynomial is expressed as the product of simpler polynomials.

   1. Factoring out the Greatest Common Factor (GCF)

   Always inspect the terms for common numerical factors and common variable powers first.

   Example: Factor $12x^4 - 18x^3 + 6x^2$.

   • The numerical GCF of 12, 18, and 6 is 6.
   • The lowest power of $x$ common to all terms is $x^2$.
   • The overall GCF is $6x^2$.

   Divide each term by $6x^2$ and write the factored expression:

   $$6x^2(2x^2 - 3x + 1)$$

   2. Factoring by Grouping

   When an expression contains four terms, group them into pairs to extract common binomial factors.

   Example: Factor $2x^3 + 6x^2 + 5x + 15$.

   1. Group into two pairs: $(2x^3 + 6x^2) + (5x + 15)$
   2. Factor the GCF from each pair: $2x^2(x + 3) + 5(x + 3)$
   3. Factor out the shared binomial factor $(x + 3)$: $$(x + 3)(2x^2 + 5)$$

   Factoring Trinomials

   A quadratic trinomial in the form $x^2 + bx + c$ can often be factored into two linear factors $(x + p)(x + q)$, where:

   $$p + q = b \quad \text{and} \quad p \cdot q = c$$

   Example: Factor $x^2 - 7x + 12$.

   Find two numbers whose product is 12 and whose sum is $-7$. The factors are $-3$ and $-4$:

   $$x^2 - 7x + 12 = (x - 3)(x - 4)$$

   Special Factoring Patterns

   Recognizing patterns streamlines polynomial operations and factoring:

   • Difference of Two Squares:
     $$a^2 - b^2 = (a - b)(a + b)$$
     Example: $9x^2 - 25 = (3x - 5)(3x + 5)$
   • Perfect Square Trinomials:
     $$a^2 + 2ab + b^2 = (a + b)^2$$
     $$a^2 - 2ab + b^2 = (a - b)^2$$
     Example: $4x^2 - 12x + 9 = (2x - 3)^2$