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Polynomial Operations

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Polynomial Operations

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Polynomial Operations

🎯 Learning Objectives

By the end of this topic, you will be able to:

Define a polynomial function and classify by degree and number of terms
Perform addition, subtraction, multiplication, and division of polynomials
Apply the Remainder Theorem and Factor Theorem
Determine the end behavior, $x$ -intercepts, multiplicities, and turning points of polynomial graphs
Sketch the graph of a polynomial function given its equation

📋 Prerequisites

Quadratic Functions & Equations — degree-2 polynomials
Algebra — exponent rules, distributive property, factoring
Coordinate geometry — plotting points, intercepts

📖 Core Content

5.1 What is a Polynomial?

5.1.1 Intuition

A polynomial is a mathematical expression built from variables and constants using only addition, subtraction, multiplication, and non-negative integer exponents. Think of it as "many terms" (poly = many, nomial = name/term). Polynomials model smooth, continuous relationships — from simple linear trends (

y = 2x + 1

) to complex curves (

y = x^5 - 3x^3 + 2x

💡 Why this matters: Polynomials are the building blocks of approximation. Taylor series (Maths 2) approximates any smooth function as a polynomial. In ML, polynomial regression captures non-linear patterns.

5.1.2 Formal Definition

A polynomial function is of the form:

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

where:

$a_n \neq 0$ is the leading coefficient
$n$ is a non-negative integer (the degree)
$a_0$ is the constant term
Each $a_k x^k$ is a term Classification by degree:

Degree	Name	Example
0	Constant	$P(x) = 5$
1	Linear	$P(x) = 2x + 3$
2	Quadratic	$P(x) = x^2 - 4x + 4$
3	Cubic	$P(x) = x^3 - x$
4	Quartic	$P(x) = x^4 - 5x^2 + 4$
$n$	$n$ th-degree	$P(x) = a_n x^n + \cdots$

Not a polynomial:

f(x) = \sqrt{x}

(fractional exponent),

f(x) = 1/x

(negative exponent),

f(x) = 2^x

(variable exponent).

5.2 Operations on Polynomials

5.2.1 Addition and Subtraction

Combine like terms (same variable and exponent). Example 1.1: Add

(3x^2 - 2x + 5) + (x^2 + 4x - 1)

. &= (3x^2 + x^2) + (-2x + 4x) + (5 - 1) \\ &= 4x^2 + 2x + 4 \end{aligned}

\\boxed{4x^2 + 2x + 4} **Example 1.2:** Subtract $(5x^3 - 2x + 7) - (3x^3 + 4x^2 - x + 2)$.

\begin{aligned} &= 5x^3 - 2x + 7 - 3x^3 - 4x^2 + x - 2 \\ &= (5x^3 - 3x^3) - 4x^2 + (-2x + x) + (7 - 2) \\ &= 2x^3 - 4x^2 - x + 5 \end{aligned}

\\boxed{2x^3 - 4x^2 - x + 5} \#### 5.2.2 Multiplication Multiply every term of one polynomial by every term of the other (distributive property). **Example 2.1 (Easy):** Multiply $(x)(x^2 - 3x + 2)$. $x \\cdot x^2 = x^3$, $x \\cdot (-3x) = -3x^2$, $x \\cdot 2 = 2x$ Result: $x^3 - 3x^2 + 2x$ \\boxed{x^3 - 3x^2 + 2x} **Example 2.2 (Medium):** Multiply $(2x + 1)(x^2 - 3x + 4)$.

\begin{aligned} &= 2x(x^2 - 3x + 4) + 1(x^2 - 3x + 4) \\ &= 2x^3 - 6x^2 + 8x + x^2 - 3x + 4 \\ &= 2x^3 - 5x^2 + 5x + 4 \end{aligned}

\\boxed{2x^3 - 5x^2 + 5x + 4} **Example 2.3 (Special products):** - $(x + a)(x - a) = x^2 - a^2$ (difference of squares) - $(x + a)^2 = x^2 + 2ax + a^2$ (perfect square) - $(x + a)(x + b) = x^2 + (a+b)x + ab$ #### 5.2.3 Division (Long Division) **Recipe for polynomial long division:** 1. Write dividend and divisor in descending powers 2. Divide leading terms: first term of quotient 3. Multiply quotient term by divisor, subtract from dividend 4. Repeat with remainder as new dividend 5. Stop when remainder degree < divisor degree **Example 3.1:** Divide $(x^3 - 2x^2 + 3x - 5) \\div (x - 2)$.

\begin{aligned} & \phantom{)} \quad x^2 + 0x + 3 \\ x - 2 &\overline{)x^3 - 2x^2 + 3x - 5} \\ &\underline{x^3 - 2x^2} \\ &\phantom{)} \quad 0x^2 + 3x \\ &\phantom{)} \quad \underline{0x^2 + 0x} \\ &\phantom{)} \qquad \quad 3x - 5 \\ &\phantom{)} \qquad \quad \underline{3x - 6} \\ &\phantom{)} \qquad \quad \qquad 1 \end{aligned}

Quotient: $x^2 + 3$, Remainder: $1$ \\boxed{x^2 + 3 + \\frac{1}{x-2}} \--- ### 5.3 Remainder Theorem #### 5.3.1 Statement If a polynomial $P(x)$ is divided by $(x - a)$, the remainder is $P(a)$. **Proof:** $P(x) = (x-a)Q(x) + R$, where $R$ is constant (since divisor is degree 1). Substitute $x = a$: $P(a) = (a-a)Q(a) + R = R$. ∎ #### 5.3.2 Worked Example Find the remainder when $P(x) = x^3 - 3x^2 + 2x + 1$ is divided by $(x - 4)$. *Step 1 — Using Remainder Theorem:* $P(4) = 64 - 48 + 8 + 1 = 25$ *Step 2 — Verify with long division (optional):* Quotient $= x^2 + x + 6$, remainder $= 25$ ✓ \\boxed{25} \--- ### 5.4 Factor Theorem #### 5.4.1 Statement $(x - a)$ is a factor of $P(x)$ **if and only if** $P(a) = 0$. This follows directly from the Remainder Theorem: if the remainder is 0, $(x - a)$ divides $P(x)$ exactly. #### 5.4.2 Worked Examples **Example 4.1:** Show that $(x - 2)$ is a factor of $P(x) = x^3 - 4x^2 + x + 6$. *Step 1:* $P(2) = 8 - 16 + 2 + 6 = 0$ *Step 2:* Since $P(2) = 0$, $(x - 2)$ is a factor. \\boxed{P(2) = 0 \\rightarrow \\text{factor}} **Example 4.2:** Factor $P(x) = x^3 - 3x^2 - 4x + 12$ completely. *Step 1 — Find a root by testing:* $P(2) = 8 - 12 - 8 + 12 = 0$. So $(x-2)$ is a factor. *Step 2 — Divide:* $(x^3 - 3x^2 - 4x + 12) \\div (x-2) = x^2 - x - 6$ *Step 3 — Factor quotient:* $x^2 - x - 6 = (x-3)(x+2)$ *Step 4 — Complete factorization:* $P(x) = (x-2)(x-3)(x+2)$ \\boxed{(x-2)(x-3)(x+2)} \--- ### 5.5 Graphs of Polynomial Functions #### 5.5.1 Key Features \`\`\`mermaid graph TD subgraph "Polynomial Graph Analysis" DEG\["Degree n"\] --> EB\["End behavior<br/>n even: same direction<br/>n odd: opposite directions"\] DEG --> TP\["Turning points<br/>At most n-1"\] ROOTS\["x-intercepts"\] --> MULT\["Multiplicity<br/>Odd: crosses<br/>Even: touches"\] COEFF\["Leading coeff an"\] --> DIR\["an>0: up-right<br/>an<0: down-right"\] end \`\`\` #### 5.5.2 End Behavior As $x \\to \\pm\\infty$, the term with highest degree dominates: | $n$ (degree) | $a_n$ (leading coeff) | $x \\to -\\infty$ | $x \\to \\infty$ | |:---:|:---:|:---:|:---:| | Even | Positive | Up | Up | | Even | Negative | Down | Down | | Odd | Positive | Down | Up | | Odd | Negative | Up | Down | #### 5.5.3 Multiplicity If $(x - r)^k$ is a factor of $P(x)$, then $r$ is a root with **multiplicity** $k$. | Multiplicity | Behavior at $x = r$ | |:---:|:---| | $k = 1$ (odd) | Graph **crosses** the $x$-axis | | $k$ odd $> 1$ | Graph **crosses** with flattening | | $k$ even | Graph **touches** (bounces off) $x$-axis | #### 5.5.4 Turning Points A polynomial of degree $n$ has at most $n-1$ turning points (local maxima/minima). #### 5.5.5 Worked Examples **Example 5.1:** Analyze $P(x) = (x+2)(x-1)^2(x-3)$. *Step 1 — Degree:* $1 + 2 + 1 = 4$ (quartic) *Step 2 — Leading coefficient:* $1 > 0$, even degree → both ends up *Step 3 — x-intercepts:* $-2$ (multiplicity 1, crosses), $1$ (mult. 2, touches), $3$ (mult. 1, crosses) *Step 4 — y-intercept:* $P(0) = (2)(1)^2(-3) = -6$ *Step 5 — Turning points:* at most $3$ \\boxed{\\text{Up/up, crosses at }-2,3,\\text{ touches at }1,\\text{ y-int }-6} **Example 5.2:** Sketch $P(x) = -x^3 + 2x^2 + x - 2$. *Step 1 — Degree:* 3 (odd), leading coeff $-1$ (negative) *Step 2 — End behavior:* $x \\to -\\infty$, up; $x \\to \\infty$, down *Step 3 — Factor:* $P(x) = -(x-1)(x+1)(x-2)$ *Step 4 — x-intercepts:* $1, -1, 2$ (all multiplicity 1, all cross) *Step 5 — y-intercept:* $P(0) = -2$ The graph starts up, crosses at $-1$, goes down then up, crosses at $1$, goes down then up, crosses at $2$, goes down. \\boxed{\\text{Up/down ends, crosses at }-1,1,2,\\text{ y-int }-2} \--- ### 5.6 Why This Matters Polynomials are universal approximators. The **Taylor series** (BSMA1003 Maths 2) shows that any smooth function can be approximated by a polynomial. In data science: - **Polynomial regression** captures non-linear trends - **Spline fitting** uses piecewise polynomials - **Activation functions** in neural networks are sometimes polynomial-ish --- ## 📐 Key Formulas — Summary Table | Concept | Formula | Notes | |:---|:---|:---| | Remainder Theorem | $P(a)$ = remainder when divided by $(x-a)$ | | | Factor Theorem | $P(a) = 0 \\iff (x-a)$ is a factor | | | Standard polynomial | $P(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_0$ | $a_n \\neq 0$ | | Degree | $n$ | Highest exponent | | End behavior (even, $a_n>0$) | Up / Up | Similar for both ends | | End behavior (odd, $a_n>0$) | Down / Up | Opposite ends | | Max turning points | $n-1$ | | | Max $x$-intercepts | $n$ | Real roots | | Division algorithm | $P(x) = (x-a)Q(x) + R$ | $R$ is constant | --- ## ⚠️ Common Pitfalls ### Pitfall 1: Forgetting to Include Missing Terms in Long Division **Mistake:** Writing $x^3 + 2x - 5$ as is when dividing, ignoring the missing $x^2$ term. **Correct:** Write $x^3 + 0x^2 + 2x - 5$ to keep columns aligned. ### Pitfall 2: Sign Errors in Subtraction **Mistake:** Subtracting $x^2 - 3x$ from $x^2 + x$ gives $4x$ (WRONG: $x^2 + x - (x^2 - 3x) = x^2 + x - x^2 + 3x = 4x$... wait that IS $4x$. Let me redo: actually the common mistake is: $x^2 + x - (x^2 - 3x) = x^2 + x - x^2 + 3x = 4x$ — that's correct. A better example: **Correct:** $(2x^3 - 5x^2) - (2x^3 + x^2) = 2x^3 - 5x^2 - 2x^3 - x^2 = -6x^2$. The mistake is forgetting to distribute the minus sign to every term of the subtracted polynomial. ### Pitfall 3: Confusing Number of Turning Points with Degree **Mistake:** A degree-4 polynomial always has exactly 3 turning points. **Correct:** At MOST 3 turning points. Could have fewer (e.g., $P(x) = x^4$ has 0 turning points). ### Pitfall 4: Ignoring Multiplicity When Sketching **Mistake:** Drawing all $x$-intercepts as "crossing" the axis. **Correct:** Check multiplicity — even multiplicity means the graph **touches** (bounces), odd means **crosses**. --- ## 📝 Practice Questions > **Q1: Add $(2x^3 - x^2 + 3) + (x^3 + 4x^2 - 2x + 1)$.** > > **Strategy Hint:** Combine like terms. > > $= 3x^3 + 3x^2 - 2x + 4$ > >

\boxed{3x^3 + 3x^2 - 2x + 4}

> **Q2: Multiply $(x + 3)(x^2 - 2x + 1)$.** > > **Strategy Hint:** Distribute each term of first polynomial. > > $= x(x^2 - 2x + 1) + 3(x^2 - 2x + 1)$ > $= x^3 - 2x^2 + x + 3x^2 - 6x + 3$ > $= x^3 + x^2 - 5x + 3$ > >

\boxed{x^3 + x^2 - 5x + 3}

> **Q3: Divide $(x^3 - 3x^2 + 5x - 2) \\div (x - 1)$ using synthetic division.** > > **Strategy Hint:** Synthetic division: coefficients 1, -3, 5, -2; evaluate at $x=1$. > > $1 \\mid 1 \\quad -3 \\quad 5 \\quad -2$ > $\\quad \\mid \\quad 1 \\quad -2 \\quad 3$ > $\\quad \\overline{1 \\quad -2 \\quad 3 \\quad 1}$ > > Result: $x^2 - 2x + 3 + \\frac{1}{x-1}$ > >

\boxed{x^2 - 2x + 3 + \frac{1}{x-1}}

> **Q4: Use the Remainder Theorem to find $P(-2)$ for $P(x) = x^3 + 2x^2 - 3x + 5$.** > > **Strategy Hint:** $P(-2)$ is the remainder when dividing by $(x+2)$. > > $P(-2) = -8 + 8 + 6 + 5 = 11$ > >

\boxed{11}

> **Q5: Is $(x - 3)$ a factor of $x^3 - 7x + 6$?** > > **Strategy Hint:** Check $P(3) = 0$? > > $P(3) = 27 - 21 + 6 = 12 \\neq 0$ → NOT a factor. > >

\boxed{\text{No, } P(3) = 12 \neq 0}

> **Q6: Find the $x$-intercepts and multiplicities for $P(x) = (x-1)^3(x+2)^2$.** > > **Strategy Hint:** Roots from each factor. > > $x = 1$ (multiplicity 3, crosses), $x = -2$ (multiplicity 2, touches). > >

\boxed{x=1 \text{ (cross), } x=-2 \text{ (touch)}}

> **Q7: Determine the end behavior of $P(x) = -2x^5 + 3x^3 - x + 7$.** > > **Strategy Hint:** Degree 5 (odd), leading coeff -2 (negative). > > As $x \\to -\\infty$: up (since odd, $a_n<0$) > As $x \\to \\infty$: down > >

\boxed{\text{Up left, down right}}

> **Q8: How many turning points can $P(x) = x^4 - 3x^2 + 2$ have at most?** > > **Strategy Hint:** Degree 4 \to at most $4-1=3$ turning points. > >

\boxed{3}

> **Q9: Factor completely: $x^3 - 4x^2 + x + 6$.** > > **Strategy Hint:** Test $x=2$: $8-16+2+6=0$, so $(x-2)$ is a factor. > > $(x^3 - 4x^2 + x + 6) \\div (x-2) = x^2 - 2x - 3 = (x-3)(x+1)$ > $P(x) = (x-2)(x-3)(x+1)$ > >

\boxed{(x-2)(x-3)(x+1)}

> **Q10: Find a cubic polynomial with roots $-2, 1, 3$ and leading coefficient $2$.** > > **Strategy Hint:** $P(x) = a(x - r_1)(x - r_2)(x - r_3)$. > > $P(x) = 2(x+2)(x-1)(x-3)$ > $= 2(x+2)(x^2 - 4x + 3)$ > $= 2(x^3 - 4x^2 + 3x + 2x^2 - 8x + 6)$ > $= 2(x^3 - 2x^2 - 5x + 6)$ > $= 2x^3 - 4x^2 - 10x + 12$ > >

\boxed{2x^3 - 4x^2 - 10x + 12}

> **Q11: The graph of $P(x)$ touches the $x$-axis at $x=2$ and crosses at $x=-1$. What can you say about the factors?** > > **Strategy Hint:** Touching \to even multiplicity. Crossing \to odd multiplicity. > > $(x-2)$ has even exponent (at least 2). $(x+1)$ has odd exponent (at least 1). > >

\boxed{(x-2)^k \text{ with } k \text{ even}, (x+1)^m \text{ with } m \text{ odd}}

> **Q12: Divide $(x^4 - 3x^3 + 2x^2 - x + 5) \\div (x^2 + 1)$.** > > **Strategy Hint:** Long division with quadratic divisor. > > $x^4 - 3x^3 + 2x^2 - x + 5$ ÷ $(x^2 + 1)$ > Quotient: $x^2 - 3x + 1$, Remainder: $2x + 4$ > >

\boxed{x^2 - 3x + 1 + \frac{2x+4}{x^2+1}}$$ --- ## 🔗 Cross-References - Previous topics: [Quadratic Functions](quadratics-introduction.md), [Quadratic Equations](quadratics-equations.md) - Next topic: [Exponential Functions](../week05/functions-exponential.md) - Across courses: BSMA1003 Maths 2 (Taylor polynomials, polynomial interpolation); BSMA1002 Stats 1 (polynomial regression) Join Discord Previous3.2 Quadratic Equations & DiscriminantNext5.1 Exponential Functions