6.3 Polynomial Operations (Addition, Subtraction & Multiplication)

Polynomial Operations: Addition, Subtraction & Multiplication

Quick Summary: A polynomial is a mathematical expression composed of terms with real coefficients and whole-number exponents. Polynomials are classified by their number of terms (monomial, binomial, trinomial) and their degree (the highest exponent). Operating on polynomials requires strict adherence to exponent rules: when adding or subtracting, combine only like terms without changing exponents; when multiplying, multiply numerical coefficients and add variable exponents ($x^m \cdot x^n = x^{m+n}$).

Polynomial arithmetic forms the backbone of algebraic modeling on the HiSET. Mastery of these operations ensures precision when expanding formulas, calculating geometric areas, and solving higher-order equations.

---

Polynomial Anatomy & Classification

A single-variable polynomial in standard form is written in descending order of exponents:

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

• Leading Term: The term with the highest power of $x$ ($a_n x^n$).
• Leading Coefficient: The numerical coefficient of the leading term ($a_n$).
• Degree: The highest exponent $n$ appearing on a variable.
• Constant Term: The term without a variable ($a_0$).

                    Anatomy of a Polynomial in Standard Form
                                   Degree = 3
                                      │
                         Leading      ▼
                       Coefficient ──► 5x³ - 4x² + 7x - 9 ◄── Constant Term
                                       ▲
                                       │
                                  Leading Term

Classification by Number of Terms & Degree

Adding & Subtracting Polynomials

What Are Like Terms?

Like terms are terms that contain the exact same variables raised to the exact same powers.

• $5x^2$ and $-8x^2$ are like terms (both have $x^2$).
• $4x^3$ and $4x^2$ are NOT like terms (exponents differ).
• $3xy^2$ and $3x^2 y$ are NOT like terms (variable powers differ).

Polynomial Addition

To add polynomials, remove parentheses and combine like terms by adding their numerical coefficients:

$$(4x^2 - 3x + 8) + (2x^2 + 7x - 5)$$ $$= (4x^2 + 2x^2) + (-3x + 7x) + (8 - 5)$$ $$= 6x^2 + 4x + 3$$

---

Polynomial Subtraction: The Negative Distribution Rule

The Cardinal Rule of Subtraction: Subtracting a polynomial is equivalent to adding its opposite. You must distribute a $-1$ across every term inside the parentheses of the subtracted polynomial before combining like terms.

$$-(a x^2 + b x + c) = -a x^2 - b x - c$$

Worked Example: Multi-Term Subtraction

Simplify: $(7x^3 - 4x^2 + 6x - 2) - (3x^3 + 5x^2 - 8x + 9)$

1. Distribute the negative sign to all terms in the second polynomial: $$7x^3 - 4x^2 + 6x - 2 - 3x^3 - 5x^2 + 8x - 9$$
2. Group like terms by degree: $$(7x^3 - 3x^3) + (-4x^2 - 5x^2) + (6x + 8x) + (-2 - 9)$$
3. Combine coefficients: $$4x^3 - 9x^2 + 14x - 11$$

   Vertical Alignment Method for Subtraction
     7x³  -  4x²  +   6x  -   2
   - (3x³  +  5x²  -   8x  +   9)  ───► Change signs and ADD
   ───────────────────────────────
     7x³  -  4x²  +   6x  -   2
   + -3x³  -  5x²  +   8x  -   9
   ───────────────────────────────
     4x³  -  9x²  +  14x  -  11

Multiplying Polynomials

Multiplying polynomials relies on the Distributive Property and the Product Rule of Exponents ($x^a \cdot x^b = x^{a+b}$).

1. Monomial $\times$ Polynomial

Multiply the outer monomial by every individual term inside:

$$-3x^2 (2x^3 - 5x + 4) = (-3x^2 \cdot 2x^3) + (-3x^2 \cdot -5x) + (-3x^2 \cdot 4)$$ $$= -6x^{2+3} + 15x^{2+1} - 12x^2 = -6x^5 + 15x^3 - 12x^2$$

2. Binomial $\times$ Binomial (The FOIL Method)

When multiplying two binomials $(a + b)(c + d)$, use the FOIL acronym:

• First: Multiply the first terms in each set of parentheses ($a \cdot c$)
• Outside: Multiply the outermost terms ($a \cdot d$)
• Inside: Multiply the innermost terms ($b \cdot c$)
• Last: Multiply the last terms in each set of parentheses ($b \cdot d$)

$$\mathbf{(a + b)(c + d) = ac + ad + bc + bd}$$

      The FOIL Multiplication Pattern
              F (First) ──► a · c
            ┌───────────┐
           (a  +  b) (c  +  d)
            │     └─────┘   │
            │    I (Inside) │
            │   ──► b · c   │
            └───────────────┘
             O (Outside) ──► a · d
             L (Last)    ──► b · d

Worked Example: FOIL with Negative Coefficients

Expand: $(3x - 4)(2x + 7)$

• First: $(3x)(2x) = 6x^2$
• Outside: $(3x)(7) = +21x$
• Inside: $(-4)(2x) = -8x$
• Last: $(-4)(7) = -28$

Combine the middle like terms ($+21x - 8x = +13x$):

$$(3x - 4)(2x + 7) = 6x^2 + 13x - 28$$

3. Binomial $\times$ Trinomial

Multiply each term of the binomial by all three terms of the trinomial, producing 6 initial terms before combining:

$$(x - 2)(3x^2 + 4x - 5)$$ $$= x(3x^2 + 4x - 5) - 2(3x^2 + 4x - 5)$$ $$= (3x^3 + 4x^2 - 5x) + (-6x^2 - 8x + 10)$$ $$= 3x^3 + (4x^2 - 6x^2) + (-5x - 8x) + 10$$ $$= 3x^3 - 2x^2 - 13x + 10$$

---

Critical Algebraic Traps & Special Expansions

Trap 1: The "Freshman's Dream" Error

Never distribute an exponent over addition or subtraction!

$$(a + b)^2 \neq a^2 + b^2 \quad \text{and} \quad (a - b)^2 \neq a^2 - b^2$$

Squaring a binomial requires writing it as a product and applying FOIL:

$$(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2$$ $$(a - b)^2 = (a - b)(a - b) = a^2 - 2ab + b^2$$

Example: $(2x - 5)^2 = (2x)^2 - 2(2x)(5) + (-5)^2 = 4x^2 - 20x + 25$. Omitting the middle term ($-20x$) is one of the most penalized errors on the HiSET.

Trap 2: Adding vs. Multiplying Variable Powers

• Addition: $3x^4 + 5x^4 = 8x^4$ (Exponents do NOT change when adding).
• Multiplication: $(3x^4)(5x^4) = 15x^{4+4} = 15x^8$ (Exponents ADD when multiplying).