# 2.1 Polynomials

## Detailed Theory: Polynomials

### **1. Basic Concepts and Definitions**

#### **1.1 What is a Polynomial?**

A polynomial is an algebraic expression consisting of variables (indeterminates) and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

#### **1.2 General Form of a Polynomial**

A polynomial in one variable $$x$$ of degree $$n$$ is written as:

$$P(x) = a\_nx^n + a\_{n-1}x^{n-1} + \cdots + a\_2x^2 + a\_1x + a\_0$$

**Where:**

$$a\_n, a\_{n-1}, \ldots, a\_1, a\_0$$ are constants called **coefficients**

$$a\_n \neq 0$$ (leading coefficient)

$$n$$ is a non-negative integer called the **degree** of the polynomial

$$a\_0$$ is the **constant term**

Each $$a\_kx^k$$ is called a **term**

#### **1.3 Standard Form**

A polynomial is in standard form when terms are arranged in **descending powers** of the variable.

**Example:**

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

is in standard form.

#### **1.4 Degree of a Polynomial**

The **degree** of a polynomial is the **highest power** of the variable with a non-zero coefficient.

**Notation:** $$\deg(P)$$ or $$\text{deg}(P)$$

**Examples:**

$$P(x) = 5x^3 - 2x^2 + x - 1$$ → Degree = $$3$$

$$Q(x) = 7$$ → Degree = $$0$$ (constant polynomial)

$$R(x) = 0$$ → Degree is undefined (zero polynomial)

#### **1.5 Classification by Degree**

1. **Constant polynomial:** Degree $$0$$ (e.g., $$P(x) = 5$$)
2. **Linear polynomial:** Degree $$1$$ (e.g., $$P(x) = 2x + 3$$)
3. **Quadratic polynomial:** Degree $$2$$ (e.g., $$P(x) = x^2 - 4x + 4$$)
4. **Cubic polynomial:** Degree $$3$$ (e.g., $$P(x) = x^3 - 6x^2 + 11x - 6$$)
5. **Quartic polynomial:** Degree $$4$$ (e.g., $$P(x) = x^4 - 10x^2 + 9$$)
6. **Quintic polynomial:** Degree $$5$$
7. **n-th degree polynomial:** Degree $$n$$

#### **1.6 Classification by Number of Terms**

1. **Monomial:** $$1$$ term (e.g., $$7x^3$$)
2. **Binomial:** $$2$$ terms (e.g., $$x^2 - 4$$)
3. **Trinomial:** $$3$$ terms (e.g., $$x^2 + 5x + 6$$)
4. **Multinomial:** More than $$3$$ terms

#### **1.7 Zero Polynomial**

The polynomial with all coefficients zero: $$P(x) = 0$$

**Special Properties:**

Its degree is **undefined** (some say $$-\infty$$ or not defined)

It's the additive identity in polynomial algebra

$$P(x) + 0 = P(x)$$ for any polynomial $$P(x)$$

***

### **2. Operations on Polynomials**

#### **2.1 Addition and Subtraction**

Add/subtract corresponding coefficients of like terms (same power).

**Example 1:** Add $$(3x^3 - 2x^2 + 5x - 1)$$ and $$(2x^3 + 4x^2 - 3x + 2)$$

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

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

$$= 5x^3 + 2x^2 + 2x + 1$$

**Example 2:** Subtract $$(4x^2 - 3x + 2)$$ from $$(2x^3 + 5x^2 - x + 1)$$

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

$$= 2x^3 + 5x^2 - x + 1 - 4x^2 + 3x - 2$$

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

$$= 2x^3 + x^2 + 2x - 1$$

#### **2.2 Multiplication**

**a) Multiplying by a Constant**

Multiply each coefficient by the constant.

**Example:**

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

$$= 6x^3 - 3x^2 + 12x - 3$$

**b) Multiplying Two Polynomials**

Use distributive property: Multiply each term of first polynomial by each term of second polynomial, then combine like terms.

**Example:** Multiply $$(2x + 3)$$ by $$(x^2 - 2x + 1)$$

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

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

$$= 2x^3 - 4x^2 + 2x + 3x^2 - 6x + 3$$

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

$$= 2x^3 - x^2 - 4x + 3$$

**Degree Property:**

$$\deg(P \cdot Q) = \deg(P) + \deg(Q)$$

#### **2.3 Division of Polynomials**

**a) Division by a Monomial**

Divide each term by the monomial.

**Example:** Divide $$(6x^4 - 9x^3 + 3x^2)$$ by $$3x^2$$

$$= \frac{6x^4 - 9x^3 + 3x^2}{3x^2}$$

$$= \frac{6x^4}{3x^2} - \frac{9x^3}{3x^2} + \frac{3x^2}{3x^2}$$

$$= 2x^2 - 3x + 1$$

**b) Long Division of Polynomials**

Similar to numerical long division.

**Example:** Divide $$(x^3 - 6x^2 + 11x - 6)$$ by $$(x - 2)$$

**Step 1:** Divide first term: $$= \frac{x^3}{x} = x^2$$

**Step 2:** Multiply: $$= x^2(x-2) = x^3 - 2x^2$$

**Step 3:** Subtract: $$= (x^3-6x^2) - (x^3-2x^2) = -4x^2$$

**Step 4:** Bring down next term: $$= -4x^2 + 11x$$

**Step 5:** Divide: $$= \frac{-4x^2}{x} = -4x$$

**Step 6:** Multiply: $$= -4x(x-2) = -4x^2 + 8x$$

**Step 7:** Subtract: $$= (-4x^2+11x) - (-4x^2+8x) = 3x$$

**Step 8:** Bring down: $$= 3x - 6$$

**Step 9:** Divide: $$= \frac{3x}{x} = 3$$

**Step 10:** Multiply: $$= 3(x-2) = 3x - 6$$

**Step 11:** Subtract: $$= (3x-6) - (3x-6) = 0$$

**Result:** Quotient = $$x^2 - 4x + 3$$, Remainder = $$0$$

**So:**

$$= x^3 - 6x^2 + 11x - 6 = (x-2)(x^2 - 4x + 3)$$

**c) Synthetic Division**

A shortcut for division by linear factors of form $$(x-c)$$.

**Example:** Divide $$(2x^3 - 7x^2 + 5x - 1)$$ by $$(x-3)$$

**Steps:**

**Step 1:** Write coefficients: $$2, -7, 5, -1$$

**Step 2:** Write $$c = 3$$ (from $$x-3$$)

**Step 3:** Bring down first coefficient: $$2$$

**Step 4:** Multiply by $$c$$ and add to next coefficient:

$$= 2 \times 3 = 6$$

$$= -7 + 6 = -1$$

**Step 5:**

$$= -1 \times 3 = -3$$

$$= 5 - 3 = 2$$

**Step 6:**

$$= 2 \times 3 = 6$$

$$= -1 + 6 = 5$$

**Step 7:** Last number is remainder ($$5$$), others are coefficients of quotient

Quotient: $$2x^2 - x + 2$$, Remainder: $$5$$

**Verification:**

$$= (2x^3 - 7x^2 + 5x - 1)$$

$$= (x-3)(2x^2 - x + 2) + 5$$

***

### **3. Polynomial Equations and Roots**

#### **3.1 Polynomial Equation**

An equation of the form $$P(x) = 0$$ where $$P(x)$$ is a polynomial.

**Example:**

$$x^3 - 6x^2 + 11x - 6 = 0$$

#### **3.2 Roots/Zeros of a Polynomial**

A number $$\alpha$$ is a **root** (or **zero**) of polynomial $$P(x)$$ if $$P(\alpha) = 0$$.

**Equivalent Statements:**

1. $$\alpha$$ is a root of $$P(x)$$
2. $$P(\alpha) = 0$$
3. $$(x - \alpha)$$ is a factor of $$P(x)$$
4. $$\alpha$$ is a solution of equation $$P(x) = 0$$

#### **3.3 Fundamental Theorem of Algebra**

Every non-constant polynomial with complex coefficients has at least one complex root.

**Consequences:**

1. A polynomial of degree $$n$$ has exactly $$n$$ roots (counting multiplicity)
2. If coefficients are real, complex roots occur in conjugate pairs

#### **3.4 Multiplicity of a Root**

If $$(x-\alpha)^k$$ is a factor of $$P(x)$$ but $$(x-\alpha)^{k+1}$$ is not, then $$\alpha$$ is a root of **multiplicity** $$k$$.

**Behavior at roots:**

Odd multiplicity: Graph crosses x-axis at root

Even multiplicity: Graph touches x-axis and turns around at root

**Example:**

$$P(x) = (x-1)^3(x+2)^2(x-3)$$

Root $$x=1$$ has multiplicity $$3$$ (odd, crosses axis)

Root $$x=-2$$ has multiplicity $$2$$ (even, touches axis)

Root $$x=3$$ has multiplicity $$1$$ (odd, crosses axis)

#### **3.5 Finding Roots**

**a) For Linear Polynomials** $$(ax + b = 0)$$

Solution: $$x = -\frac{b}{a}$$

**b) For Quadratic Polynomials** $$(ax^2 + bx + c = 0)$$

**Quadratic Formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

**Discriminant** $$(\Delta)$$**:** $$\Delta = b^2 - 4ac$$

If $$\Delta > 0$$: Two distinct real roots

If $$\Delta = 0$$: One real root (double root)

If $$\Delta < 0$$: Two complex conjugate roots

**c) For Cubic Polynomials**

General cubic: $$ax^3 + bx^2 + cx + d = 0$$

**Special Cases:**

1. **Factorable:** Try integer factors of constant term
2. **Sum/difference of cubes:** $$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$$
3. **Cardano's formula** (general solution, but complex)

**Example:** Solve $$x^3 - 6x^2 + 11x - 6 = 0$$

Try factors of $$6$$: $$\pm1, \pm2, \pm3, \pm6$$

**Check:** $$P(1) = 1 - 6 + 11 - 6 = 0 \quad \checkmark$$

So $$(x-1)$$ is a factor.

Divide: $$= \frac{x^3 - 6x^2 + 11x - 6}{x-1} = x^2 - 5x + 6$$

Solve quadratic: $$x^2 - 5x + 6 = 0 \rightarrow (x-2)(x-3) = 0$$

Roots: $$x = 1, 2, 3$$

**d) For Quartic Polynomials**

General quartic: $$ax^4 + bx^3 + cx^2 + dx + e = 0$$

**Special Cases:**

1. **Biquadratic:** $$ax^4 + bx^2 + c = 0$$ (substitute $$y = x^2$$)
2. **Factorable:** Try rational roots
3. **Ferrari's method** (general solution, complex)

**Example:** Solve $$x^4 - 5x^2 + 4 = 0$$

Let $$y = x^2$$: $$y^2 - 5y + 4 = 0$$

Solve: $$(y-1)(y-4) = 0 \rightarrow y=1 \text{ or } y=4$$

So $$x^2 = 1$$ or $$x^2 = 4$$

Roots: $$x = \pm1, \pm2$$

#### **3.6 Rational Root Theorem**

For polynomial $$P(x) = a\_nx^n + a\_{n-1}x^{n-1} + \cdots + a\_0$$ with integer coefficients:

If $$p/q$$ (in lowest terms) is a rational root, then:

$$p$$ divides constant term $$a\_0$$

$$q$$ divides leading coefficient $$a\_n$$

**Example:** Find rational roots of $$P(x) = 2x^3 - 3x^2 - 8x + 12$$

**Possible** $$p$$**:** factors of $$12$$: $$\pm1, \pm2, \pm3, \pm4, \pm6, \pm12$$

**Possible** $$q$$**:** factors of $$2$$: $$\pm1, \pm2$$

**Possible** $$p/q$$**:** $$\pm1, \pm2, \pm3, \pm4, \pm6, \pm12, \pm\frac{1}{2}, \pm\frac{3}{2}$$

**Test:** $$P(2) = 2(8) - 3(4) - 8(2) + 12 = 16 - 12 - 16 + 12 = 0 \quad \checkmark$$

So $$x=2$$ is a root.

#### **3.7 Relationship Between Roots and Coefficients**

**For Quadratic:** $$ax^2 + bx + c = 0$$ **with roots** $$\alpha, \beta$$

1. Sum of roots: $$\alpha + \beta = -\frac{b}{a}$$
2. Product of roots: $$\alpha\beta = \frac{c}{a}$$
3. Quadratic with given roots: $$x^2 - (\alpha+\beta)x + \alpha\beta = 0$$

**For Cubic:** $$ax^3 + bx^2 + cx + d = 0$$ **with roots** $$\alpha, \beta, \gamma$$

1. Sum: $$\alpha + \beta + \gamma = -\frac{b}{a}$$
2. Sum of products taken two at a time: $$\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$$
3. Product: $$\alpha\beta\gamma = -\frac{d}{a}$$

**For Quartic:** $$ax^4 + bx^3 + cx^2 + dx + e = 0$$ **with roots** $$\alpha, \beta, \gamma, \delta$$

1. Sum: $$\alpha + \beta + \gamma + \delta = -\frac{b}{a}$$
2. Sum of products taken two at a time: $$\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta = \frac{c}{a}$$
3. Sum of products taken three at a time: $$\alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta = -\frac{d}{a}$$
4. Product: $$\alpha\beta\gamma\delta = \frac{e}{a}$$

**Example:** For cubic $$x^3 - 6x^2 + 11x - 6 = 0$$ with roots $$1, 2, 3$$:

**Sum:** $$1+2+3 = 6 = -(-6)/1 \quad \checkmark$$

**Sum of products:** $$1\cdot2 + 2\cdot3 + 3\cdot1 = 2+6+3 = 11 = 11/1 \quad \checkmark$$

**Product:** $$1\cdot2\cdot3 = 6 = -(-6)/1 = 6 \quad \checkmark$$

#### **3.8 Formation of Polynomial from Roots**

If $$\alpha\_1, \alpha\_2, \ldots, \alpha\_n$$ are roots, then:

$$P(x) = a(x-\alpha\_1)(x-\alpha\_2)\cdots(x-\alpha\_n)$$

where $$a$$ is leading coefficient.

**Example:** Form cubic with roots $$1, -2, 3$$ and leading coefficient $$2$$:

$$= P(x) = 2(x-1)(x+2)(x-3)$$

Expanding:

$$= 2[(x-1)(x+2)](https://nec-license.gitbook.io/books/agriculture-master/section-a-mathematics/x-3)$$

$$= 2(x^2+x-2)(x-3)$$

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

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

$$= 2x^3 - 4x^2 - 10x + 12$$

***

### **4. Special Polynomials and Identities**

#### **4.1 Difference of Squares**

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

**Examples:**

1. $$x^2 - 9 = (x-3)(x+3)$$
2. $$4x^2 - 25 = (2x-5)(2x+5)$$
3. $$x^4 - 16 = (x^2-4)(x^2+4) = (x-2)(x+2)(x^2+4)$$

#### **4.2 Perfect Square Trinomials**

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

**Examples:**

1. $$x^2 + 6x + 9 = (x+3)^2$$
2. $$4x^2 - 12x + 9 = (2x-3)^2$$

#### **4.3 Sum and Difference of Cubes**

1. $$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$$
2. $$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$$

**Examples:**

1. $$x^3 + 8 = (x+2)(x^2 - 2x + 4)$$
2. $$27x^3 - 1 = (3x-1)(9x^2 + 3x + 1)$$

#### **4.4 Square of a Trinomial**

$$(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$$

**Example:**

$$(x+y+1)^2 = x^2 + y^2 + 1 + 2xy + 2y + 2x$$

#### **4.5 Binomial Theorem for Polynomial Expansion**

$$(a+b)^n = \sum\_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

where $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

**Examples:**

1. **For** $$(x+2)^3$$**:**

   $$= (x+2)^3$$

   $$= \binom{3}{0}x^3 + \binom{3}{1}x^2(2) + \binom{3}{2}x(2^2) + \binom{3}{3}(2^3)$$

   $$= 1\cdot x^3 + 3\cdot 2x^2 + 3\cdot 4x + 1\cdot 8$$

   $$= x^3 + 6x^2 + 12x + 8$$
2. **For** $$(2x-1)^4$$**:**

   $$= (2x-1)^4$$

   $$= \binom{4}{0}(2x)^4 + \binom{4}{1}(2x)^3(-1) + \binom{4}{2}(2x)^2(-1)^2 + \binom{4}{3}(2x)(-1)^3 + \binom{4}{4}(-1)^4$$

   $$= 16x^4 - 32x^3 + 24x^2 - 8x + 1$$

***

### **5. Remainder Theorem and Factor Theorem**

#### **5.1 Remainder Theorem**

If polynomial $$P(x)$$ is divided by $$(x-c)$$, then the remainder is $$P(c)$$.

**Proof:**

By division algorithm: $$P(x) = (x-c)Q(x) + R$$

Put $$x = c$$: $$P(c) = (c-c)Q(c) + R = R$$

**Example:** Find remainder when $$P(x) = x^3 - 2x^2 + 3x - 1$$ is divided by $$(x-2)$$

Remainder $$= P(2) = 8 - 8 + 6 - 1 = 5$$

#### **5.2 Factor Theorem**

$$(x-c)$$ is a factor of $$P(x)$$ if and only if $$P(c) = 0$$.

**Proof:**

From Remainder Theorem: $$P(c) =$$ remainder when divided by $$(x-c)$$

If $$P(c) = 0$$, remainder $$= 0$$, so $$(x-c)$$ is a factor.

**Example:** Check if $$(x-3)$$ is factor of $$P(x) = x^3 - 6x^2 + 11x - 6$$

$$P(3) = 27 - 54 + 33 - 6 = 0 \quad \checkmark$$

So $$(x-3)$$ is a factor.

#### **5.3 Applications**

**a) Finding Roots**

Use Factor Theorem to find factors, then find roots.

**Example:** Find roots of $$P(x) = x^3 - 3x^2 - x + 3$$

Try factors of $$3$$: $$\pm1, \pm3$$

**Check:** $$P(1) = 1 - 3 - 1 + 3 = 0 \quad \checkmark$$

So $$(x-1)$$ is factor.

Divide: $$P(x) = (x-1)(x^2 - 2x - 3)$$

Solve quadratic: $$x^2 - 2x - 3 = 0 \rightarrow (x-3)(x+1) = 0$$

Roots: $$x = 1, 3, -1$$

**b) Finding Polynomial with Given Roots**

**Example:** Find quadratic with roots $$2$$ and $$-3$$

Factors: $$(x-2)$$ and $$(x+3)$$

Polynomial: $$(x-2)(x+3) = x^2 + x - 6$$

**c) Determining Unknown Coefficients**

**Example:** Find $$k$$ if $$(x-2)$$ is factor of $$P(x) = x^3 + kx^2 - 4x - 8$$

Since $$(x-2)$$ is factor, $$P(2) = 0$$

$$P(2) = 8 + 4k - 8 - 8 = 4k - 8 = 0$$

So $$k = 2$$

***

### **6. Graphs of Polynomial Functions**

#### **6.1 General Shape**

A polynomial of degree $$n$$:

Has at most $$(n-1)$$ turning points

Has at most $$n$$ x-intercepts (real roots)

End behavior depends on degree and leading coefficient

#### **6.2 End Behavior**

For $$P(x) = a\_nx^n + \cdots$$:

1. **n odd,** $$a\_n > 0$$**:** As $$x \to -\infty$$, $$P(x) \to -\infty$$; As $$x \to \infty$$, $$P(x) \to \infty$$
2. **n odd,** $$a\_n < 0$$**:** As $$x \to -\infty$$, $$P(x) \to \infty$$; As $$x \to \infty$$, $$P(x) \to -\infty$$
3. **n even,** $$a\_n > 0$$**:** As $$x \to \pm\infty$$, $$P(x) \to \infty$$
4. **n even,** $$a\_n < 0$$**:** As $$x \to \pm\infty$$, $$P(x) \to -\infty$$

**Mnemonic:** "Positive leading coefficient: Right end goes up"

#### **6.3 Key Features**

**a) Linear** $$(n=1)$$

Graph: Straight line

Equation: $$y = mx + c$$

Slope $$= m$$

y-intercept $$= c$$

**b) Quadratic** $$(n=2)$$

Graph: Parabola

Equation: $$y = ax^2 + bx + c$$

**Vertex form:** $$y = a(x-h)^2 + k$$ where vertex $$= (h,k)$$

**Vertex coordinates:** $$h = -\frac{b}{2a}, \quad k = f(h)$$

**Properties:**

Opens upward if $$a > 0$$, downward if $$a < 0$$

Axis of symmetry: $$x = h$$

y-intercept: $$c$$

x-intercepts: roots of equation

**c) Cubic** $$(n=3)$$

Graph: S-shaped curve

General shape depends on sign of leading coefficient.

**Special Case:** Cubic with three real distinct roots

Passes through x-axis at each root

Has two turning points (local max and min)

**Example:**

$$y = x^3 - 6x^2 + 11x - 6 = (x-1)(x-2)(x-3)$$

Roots at $$x=1, 2, 3$$

Two turning points between roots

***

### **7. Inequalities Involving Polynomials**

#### **7.1 Polynomial Inequalities**

To solve $$P(x) > 0$$ or $$P(x) < 0$$:

**Method:**

**Step 1:** Find all real roots of $$P(x) = 0$$

**Step 2:** Plot roots on number line

**Step 3:** Test sign in each interval

**Step 4:** Include/exclude endpoints based on inequality $$(>, \geq, <, \leq)$$

**Example:** Solve $$x^3 - 6x^2 + 11x - 6 > 0$$

Factor: $$(x-1)(x-2)(x-3) > 0$$

Roots: $$x = 1, 2, 3$$

**Test intervals:**

**Interval 1:** $$x < 1 \quad$$ Test $$x=0$$: $$(-)(-)(-) = -$$ (negative)

**Interval 2:** $$1 < x < 2 \quad$$ Test $$x=1.5$$: $$(+)(-)(-) = +$$ (positive)

**Interval 3:** $$2 < x < 3 \quad$$ Test $$x=2.5$$: $$(+)(+)(-) = -$$ (negative)

**Interval 4:** $$x > 3 \quad$$ Test $$x=4$$: $$(+)(+)(+) = +$$ (positive)

**Solution:** $$x \in (1, 2) \cup (3, \infty)$$

#### **7.2 Rational Inequalities**

Solve $$\frac{P(x)}{Q(x)} > 0$$ or similar:

**Method:**

**Step 1:** Find roots of $$P(x)=0$$ and $$Q(x)=0$$

**Step 2:** Plot all on number line (roots of $$Q(x)$$ are excluded)

**Step 3:** Test sign in each interval

**Example:** Solve $$\frac{x^2 - 4}{x - 1} > 0$$

Factor: $$\frac{(x-2)(x+2)}{x-1} > 0$$

Critical points: $$x = -2, 1, 2$$

**Test intervals:**

**Interval 1:** $$x < -2 \quad$$ Test $$x=-3$$: $$(-)(-)/(-) = (-)$$ (negative)

**Interval 2:** $$-2 < x < 1 \quad$$ Test $$x=0$$: $$(-)(+)/(-) = (+)$$ (positive)

**Interval 3:** $$1 < x < 2 \quad$$ Test $$x=1.5$$: $$(-)(+)/(+) = (-)$$ (negative)

**Interval 4:** $$x > 2 \quad$$ Test $$x=3$$: $$(+)(+)/(+) = (+)$$ (positive)

**Solution:** $$x \in (-2, 1) \cup (2, \infty)$$

***

### **8. Solved Examples**

#### **Example 1:** Complete Factorization

Factor completely: $$P(x) = x^4 - 5x^2 + 4$$

**Solution:**

**Step 1:** Let $$y = x^2$$: $$= y^2 - 5y + 4 = (y-1)(y-4)$$

**Step 2:** So $$P(x) = (x^2-1)(x^2-4)$$

**Step 3:** $$= P(x) = (x-1)(x+1)(x-2)(x+2)$$

#### **Example 2:** Find Polynomial with Given Conditions

Find cubic polynomial with roots $$1, 2, 3$$ and $$P(0) = 12$$.

**Solution:**

**Step 1:** General form: $$P(x) = a(x-1)(x-2)(x-3)$$

**Step 2:** Given $$P(0) = 12$$: $$a(-1)(-2)(-3) = 12$$

**Step 3:** $$-6a = 12 \rightarrow a = -2$$

**Step 4:** So $$P(x) = -2(x-1)(x-2)(x-3) = -2(x^3 - 6x^2 + 11x - 6)$$

**Step 5:** $$= P(x) = -2x^3 + 12x^2 - 22x + 12$$

#### **Example 3:** Synthetic Division Application

Divide $$P(x) = 3x^4 - 4x^3 + 2x^2 - 5x + 1$$ by $$(x-2)$$

**Solution using synthetic division:**

**Step 1:** Coefficients: $$3, -4, 2, -5, 1$$ $$c = 2$$

**Step 2:** Bring down $$3$$

**Step 3:** $$= 3\times2=6$$ $$= -4+6=2$$

**Step 4:** $$= 2\times2=4$$ $$= 2+4=6$$

**Step 5:** $$= 6\times2=12$$ $$= -5+12=7$$

**Step 6:** $$= 7\times2=14$$ $$= 1+14=15$$

**Result:**

Quotient: $$3x^3 + 2x^2 + 6x + 7$$

Remainder: $$15$$

So: $$= 3x^4 - 4x^3 + 2x^2 - 5x + 1$$

$$= (x-2)(3x^3+2x^2+6x+7) + 15$$

#### **Example 4:** Roots and Coefficients Relationship

If $$\alpha, \beta$$ are roots of $$x^2 - 3x + 2 = 0$$, find $$\alpha^3 + \beta^3$$.

**Solution:**

**Step 1:** From equation: $$\alpha+\beta = 3$$, $$\alpha\beta = 2$$

**Step 2:** We know: $$\alpha^3 + \beta^3 = (\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta)$$

**Step 3:** $$= 3^3 - 3\times2\times3 = 27 - 18 = 9$$

#### **Example 5:** Inequality Solution

Solve: $$(x-1)(x-2)^2(x-3)^3 > 0$$

**Solution:**

**Step 1:** Roots:

$$x=1$$ (multiplicity $$1$$)

$$x=2$$ (multiplicity $$2$$)

$$x=3$$ (multiplicity $$3$$)

**Step 2:** Sign chart:

**Interval 1:** $$x < 1 \quad$$ Test $$x=0$$: $$(-)(+)(-) = +$$ (positive)

**Interval 2:** $$1 < x < 2 \quad$$ Test $$x=1.5$$: $$(+)(+)(-) = -$$ (negative)

**Interval 3:** $$2 < x < 3 \quad$$ Test $$x=2.5$$: $$(+)(+)(-) = -$$ (negative)

**Interval 4:** $$x > 3 \quad$$ Test $$x=4$$: $$(+)(+)(+) = +$$ (positive)

**Step 3:** Since inequality is $$> 0$$, and:

At $$x=1$$: sign changes (odd multiplicity)

At $$x=2$$: sign doesn't change (even multiplicity)

At $$x=3$$: sign changes (odd multiplicity)

**Solution:** $$x \in (-\infty, 1) \cup (3, \infty)$$

***

### **9. Practice Tips for Exams**

1. **Factorization First:** Always try to factor polynomials first
2. **Rational Root Theorem:** Use for finding possible rational roots
3. **Synthetic Division:** Faster than long division for $$(x-c)$$
4. **Sum/Product of Roots:** Quick way to check answers
5. **End Behavior:** Helps sketch graphs quickly
6. **Sign Charts:** Essential for solving inequalities
7. **Multiplicity:** Determines graph behavior at roots
8. **Practice Identities:** Memorize common expansions/factorizations

This comprehensive theory covers all aspects of polynomials with detailed explanations and examples, providing complete preparation for the entrance examination.
