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 $P(x)$

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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:

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

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

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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:

$= 2[(x-1)(x+2)](x-3)$

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

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

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

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

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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 $(2x-1)^4$:

   $= (2x-1)^4$

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

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

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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 $k = 2$

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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

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

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

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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.