Polynomials and Division

Subject: Additional Mathematics
Topic: 3
Cambridge Code: 4037 / 0606

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Introduction to Polynomials

Polynomial - Expression with terms containing powers of a variable with non-negative integer exponents

Standard Form

$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

where $a_n \neq 0$ and $n$ is the degree

Examples

• $P(x) = 2x^3 - 5x^2 + 3x - 7$ (degree 3, cubic)
• $Q(x) = x^4 + 2x^2 - 1$ (degree 4, quartic)
• $R(x) = 5x + 2$ (degree 1, linear)

Degree

Degree - Highest power of the variable

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Polynomial Division

Long Division Method

Used to divide $P(x)$ by divisor $D(x)$

Form: $P(x) = D(x) \times Q(x) + R(x)$

where:

• $P(x)$ is the dividend
• $D(x)$ is the divisor
• $Q(x)$ is the quotient
• $R(x)$ is the remainder

Example: Divide $x^3 - 5x^2 + 8x - 4$ by $x - 1$

              x² - 4x + 4
         ___________________
x - 1  | x³ - 5x² + 8x - 4
         x³ - x²
         ___________
           -4x² + 8x
           -4x² + 4x
           __________
                4x - 4
                4x - 4
                _______
                   0

Result: $\frac{x^3 - 5x^2 + 8x - 4}{x-1} = x^2 - 4x + 4$

Synthetic Division

Faster method for dividing by linear divisors $(x - a)$

Steps:

1. List coefficients of dividend
2. Use divisor value $a$
3. Apply synthetic division algorithm

Example: Divide $x^3 - 5x^2 + 8x - 4$ by $x - 1$

1 |  1  -5   8  -4
  |      1  -4   4
  |_________________
     1  -4   4   0

Result: Quotient $x^2 - 4x + 4$, remainder $0$

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Remainder Theorem

Remainder Theorem - When polynomial $P(x)$ is divided by $(x-a)$, the remainder is $P(a)$

$P(x) = (x-a) \times Q(x) + P(a)$

Finding Remainder

To find remainder when $P(x)$ is divided by $(x-a)$:

Simply evaluate $P(a)$

Example

Find remainder when $P(x) = x^3 - 5x^2 + 8x - 4$ is divided by $(x-1)$

$P(1) = 1 - 5 + 8 - 4 = 0$

Remainder is $0$ ✓

Another Example

Find remainder when $P(x) = 2x^3 + 3x^2 - 5x + 1$ is divided by $(x+2)$

$P(-2) = 2(-8) + 3(4) - 5(-2) + 1$ $= -16 + 12 + 10 + 1 = 7$

Remainder is $7$

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Factor Theorem

Factor Theorem - $(x-a)$ is a factor of $P(x)$ if and only if $P(a) = 0$

Application

If $P(a) = 0$, then $(x-a)$ is a factor and: $P(x) = (x-a) \times Q(x)$

Finding Factors

Steps:

1. List possible factors using rational root theorem
2. Test each: if $P(a) = 0$, then $(x-a)$ is a factor
3. Use polynomial division to find quotient $Q(x)$
4. Repeat with $Q(x)$ if degree > 1

Example

Factorize $P(x) = x^3 - 5x^2 + 8x - 4$

Test $x = 1$: $P(1) = 1 - 5 + 8 - 4 = 0$ ✓

So $(x-1)$ is a factor

Using division: $P(x) = (x-1)(x^2-4x+4)$

Factorizing quadratic: $x^2 - 4x + 4 = (x-2)^2$

Complete factorization: $P(x) = (x-1)(x-2)^2$

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Rational Root Theorem

Rational Root Theorem - Possible rational roots of $P(x) = a_nx^n + \cdots + a_0$ are:

$\pm \frac{\text{factors of } a_0}{\text{factors of } a_n}$

Example

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

• Possible rational roots: $\pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}$

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Key Points to Remember

1. Polynomial division similar to long division
2. Remainder theorem: remainder = $P(a)$ when dividing by $(x-a)$
3. Factor theorem: $P(a) = 0$ means $(x-a)$ is a factor
4. Always check remainder or use factor theorem
5. Rational root theorem limits possible roots to test

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Worked Examples

Example 1: Polynomial Division

Divide $2x^3 + x^2 - 7x - 6$ by $x + 2$

Using synthetic division with $a = -2$:

-2 |  2   1  -7  -6
   |     -4   6   2
   |_________________
      2  -3  -1   0

Result: Quotient $2x^2 - 3x - 1$, remainder $0$

Example 2: Remainder Theorem

Find remainder when $P(x) = x^4 + 2x^3 - 3x + 5$ is divided by $(x-2)$

$P(2) = 16 + 16 - 6 + 5 = 31$

Remainder is $31$

Example 3: Factor Theorem

Show that $(x+1)$ is a factor of $P(x) = x^3 + 2x^2 - x - 2$

$P(-1) = -1 + 2 + 1 - 2 = 0$ ✓

Yes, $(x+1)$ is a factor

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Practice Questions

1. Divide $x^4 - 3x^2 + 5x - 2$ by:

   • $x - 1$
   • $x + 2$

2. Use remainder theorem to find remainder when $P(x) = 2x^3 - 5x + 3$ is divided by $(x-1)$ and $(x+2)$

3. Factorize completely:

   • $x^3 - 6x^2 + 11x - 6$
   • $2x^3 + 3x^2 - 8x - 12$

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Revision Tips

• Remainder theorem saves time finding remainders
• Factor theorem identifies roots directly
• Synthetic division faster for linear divisors
• Always check factors by testing with factor theorem