1. Polynomials

Definition:
A polynomial is an algebraic expression with terms having non-negative integer powers and real coefficients.

Standard form:
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

aₙ, aₙ₋₁, ..., a₀ → coefficients

n → degree of polynomial (highest power)

aₙxⁿ → leading term

Characteristics:

Powers of variables are non-negative integers.

Can have one or more variables.

Coefficients are real numbers.

Degree = highest power of variable.

Examples:

4x + 5y → Polynomial (degree 1)

3y³ – 1/2 y² + 5y – 7 → Polynomial (degree 3)

7y⁵ – 2y^(3/2) + 7√y – 6 → Not a polynomial (fractional powers)

2. Division of Polynomials

Division Algorithm:
For polynomials P(x) and D(x):
P(x) = D(x) × Q(x) + R(x)

Q(x) → Quotient

R(x) → Remainder

Methods:

a) Long Division
Example: Divide x² – 5x + 6 by x – 2

Step 1: Divide first term: x² ÷ x = x
Step 2: Multiply divisor: x × (x – 2) = x² – 2x
Step 3: Subtract: (x² – 5x + 6) – (x² – 2x) = -3x + 6
Step 4: Divide -3x ÷ x = -3
Step 5: Multiply divisor: -3 × (x – 2) = -3x + 6
Step 6: Subtract remainder: (-3x + 6) – (-3x + 6) = 0

Answer: Quotient = x – 3, Remainder = 0

b) Synthetic Division (for divisor x – a)
Example: Divide x⁴ – x³ – 3x² – 2x + 5 by x – 2

Step 1: Write coefficients: 1, -1, -3, -2, 5
Step 2: Bring down 1 → 1
Step 3: Multiply 1 × 2 = 2, add to next: -1 + 2 = 1
Step 4: Multiply 1 × 2 = 2, add to next: -3 + 2 = -1
Step 5: Multiply -1 × 2 = -2, add to next: -2 + (-2) = -4
Step 6: Multiply -4 × 2 = -8, add to next: 5 + (-8) = -3

Answer: Quotient = x³ + x² – x – 4, Remainder = -3

3. Remainder Theorem

Remainder when f(x) is divided by x – a = f(a)

Example: Divide f(x) = 2x³ – 3x² + 4x + 7 by x – 1

Step 1: Substitute x = 1 → f(1) = 2 – 3 + 4 + 7 = 10

Answer: Remainder = 10

4. Factor Theorem

If f(a) = 0 → x – a is a factor of f(x)

Example: Factor x³ + 6x² + 11x + 6

Step 1: Check values: f(-1) = 0 → x + 1 is a factor
Step 2: Divide polynomial by x + 1 → Quotient: x² + 5x + 6
Step 3: Factor quotient: x² + 5x + 6 = (x + 2)(x + 3)

Answer: Factors: (x + 1)(x + 2)(x + 3)

5. Roots or Zeros of a Polynomial

Roots = values of x for which f(x) = 0

Example: Solve 6x³ – 13x² + x + 2 = 0

Step 1: Factor by trial: x – 2 is a factor
Step 2: Divide polynomial by x – 2 → 6x² – x – 1
Step 3: Factor 6x² – x – 1 → (2x – 1)(3x + 1)

Answer: Roots = 2, 1/2, -1/3

6. Graphs of Polynomials

Roots correspond to x-intercepts

Degree determines shape and number of turning points

Even degree → parabola-like shape, Odd degree → S-shaped

7. Practice Problems with Solutions

Q1. Long Division: Divide x² – 10x + 21 by x – 3

Step 1: x² ÷ x = x → Multiply: x × (x – 3) = x² – 3x
Step 2: Subtract: (x² – 10x + 21) – (x² – 3x) = -7x + 21
Step 3: -7x ÷ x = -7 → Multiply: -7 × (x – 3) = -7x + 21
Step 4: Subtract: (-7x + 21) – (-7x + 21) = 0

Answer: Quotient = x – 7, Remainder = 0

Q2. Remainder Theorem: Find remainder when 4x³ + 2x² – 4x + 3 is divided by 2x + 3

Step 1: Divide by 2x + 3 → x = -3/2
Step 2: f(-3/2) = 4(-27/8) + 2(9/4) – 4(-3/2) + 3 = -27/2 + 9/2 + 6 + 3 = 0

Answer: Remainder = 0

Q3. Factor Theorem: Factor x³ – 6x² + 11x – 6

Step 1: Test factors: f(1) = 0 → x – 1 is factor
Step 2: Divide polynomial by x – 1 → x² – 5x + 6
Step 3: Factor x² – 5x + 6 → (x – 2)(x – 3)

Answer: Factors = (x – 1)(x – 2)(x – 3)

Q4. Roots: Solve 2x³ – 5x² – 4x + 12 = 0

Step 1: Try rational roots → x = 2 is root
Step 2: Divide by x – 2 → 2x² – x – 6
Step 3: Factor 2x² – x – 6 → (2x + 3)(x – 2)

Answer: Roots = 2, 2, -3/2

1. Long Division of Polynomials

Example 1: Divide x³ + 2x² – 5x – 6 by x + 3

Step 1: Divide first term: x³ ÷ x = x²
Step 2: Multiply divisor: x² × (x + 3) = x³ + 3x²
Step 3: Subtract: (x³ + 2x² – 5x – 6) – (x³ + 3x²) = -x² – 5x – 6
Step 4: Divide -x² ÷ x = -x
Step 5: Multiply: -x × (x + 3) = -x² – 3x
Step 6: Subtract: (-x² – 5x – 6) – (-x² – 3x) = -2x – 6
Step 7: Divide -2x ÷ x = -2
Step 8: Multiply: -2 × (x + 3) = -2x – 6
Step 9: Subtract remainder: (-2x – 6) – (-2x – 6) = 0

Answer: Quotient = x² – x – 2, Remainder = 0

Example 2: Divide 2x³ – 5x² + x + 2 by x – 2

Step 1: 2x³ ÷ x = 2x² → Multiply: 2x² × (x – 2) = 2x³ – 4x²
Step 2: Subtract: (2x³ – 5x² + x + 2) – (2x³ – 4x²) = -x² + x + 2
Step 3: -x² ÷ x = -x → Multiply: -x(x – 2) = -x² + 2x
Step 4: Subtract: (-x² + x + 2) – (-x² + 2x) = -x + 2
Step 5: -x ÷ x = -1 → Multiply: -1 × (x – 2) = -x + 2
Step 6: Subtract remainder: (-x + 2) – (-x + 2) = 0

Answer: Quotient = 2x² – x – 1, Remainder = 0

2. Synthetic Division

Example 1: Divide x³ – 6x² + 11x – 6 by x – 1

Step 1: Write coefficients: 1, -6, 11, -6
Step 2: Bring down 1 → 1
Step 3: Multiply 1 × 1 = 1 → Add: -6 + 1 = -5
Step 4: Multiply -5 × 1 = -5 → Add: 11 + (-5) = 6
Step 5: Multiply 6 × 1 = 6 → Add: -6 + 6 = 0

Answer: Quotient = x² – 5x + 6, Remainder = 0

3. Remainder Theorem

Example 1: Find remainder when f(x) = 3x³ – 2x² + 5x – 4 is divided by x – 2

Step 1: Substitute x = 2 → f(2) = 3(8) – 2(4) + 5(2) – 4
Step 2: Calculate: 24 – 8 + 10 – 4 = 22

Answer: Remainder = 22

Example 2: f(x) = 2x³ + 3x² – x + 5, divide by x + 1

Step 1: Substitute x = -1 → f(-1) = 2(-1)³ + 3(-1)² – (-1) + 5
Step 2: Calculate: -2 + 3 + 1 + 5 = 7

Answer: Remainder = 7

4. Factor Theorem

Example 1: Factor x³ – 4x² – 7x + 10

Step 1: Check possible roots: ±1, ±2, ±5, ±10
Step 2: Test x = 2 → f(2) = 8 – 16 – 14 + 10 = -12 → Not a root
Step 3: Test x = 1 → f(1) = 1 – 4 – 7 + 10 = 0 → x – 1 is factor
Step 4: Divide by x – 1 → Quotient: x² – 3x – 10
Step 5: Factor x² – 3x – 10 → (x – 5)(x + 2)

Answer: Factors = (x – 1)(x – 5)(x + 2)

Example 2: Factor 2x³ – 5x² – 4x + 3

Step 1: Test x = 1 → 2 – 5 – 4 + 3 = -4 → Not root
Step 2: Test x = 3 → 54 – 45 – 12 + 3 = 0 → x – 3 is factor
Step 3: Divide by x – 3 → Quotient: 2x² + x – 1
Step 4: Factor 2x² + x – 1 → (2x – 1)(x + 1)

Answer: Factors = (x – 3)(2x – 1)(x + 1)

5. Finding Roots

Example 1: Solve x³ – 7x² + 10x + 8 = 0

Step 1: Test x = 1 → 1 – 7 + 10 + 8 = 12 → Not root
Step 2: Test x = -1 → -1 – 7 – 10 + 8 = -10 → Not root
Step 3: Test x = 2 → 8 – 28 + 20 + 8 = 8 → Not root
Step 4: Test x = 4 → 64 – 112 + 40 + 8 = 0 → x – 4 is factor
Step 5: Divide by x – 4 → Quotient: x² – 3x – 2
Step 6: Factor x² – 3x – 2 → (x – 2)(x + 1)

Answer: Roots = 4, 2, -1

Example 2: Solve 3x³ – x² – 8x + 4 = 0

Step 1: Test x = 1 → 3 – 1 – 8 + 4 = -2 → Not root
Step 2: Test x = 2 → 24 – 4 – 16 + 4 = 8 → Not root
Step 3: Test x = -1 → -3 – 1 + 8 + 4 = 8 → Not root
Step 4: Test x = 4 → 192 – 16 – 32 + 4 = 148 → Not root
Step 5: Test x = -0.5 → Check manually → x + 0.5 is factor
Step 6: Divide → Remaining quadratic → Solve by factorization or formula

Answer: Roots found step by step after division

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