Cubic Polynomials Revision Notes for Grade 12 NSC Matric Mathematics

Cubic Polynomials

Introduction to polynomial division

Before we explore cubic polynomials, we need to understand how division works with polynomials. Just like with ordinary numbers, we can divide one polynomial by another to get a quotient and remainder.

The division algorithm

When we divide integers, we follow a basic pattern. For example, when we divide 11 by 2, we get 5 with remainder 1.

This can be written as: $11 = 2 \times 5 + 1$

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The Division Algorithm for Integers

In general terms, when we divide an integer $a$ by an integer $b$ , we get:

$a = b \times q + r$

where:

$a$ is the dividend (the number being divided)
$b$ is the divisor (the number we're dividing by)
$q$ is the quotient (the result)
$r$ is the remainder (what's left over)
$b \neq 0$ and $0 \leq r < b$

Extension to polynomials

This same principle applies to polynomials. The division algorithm extends beautifully from integers to polynomial expressions.

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Polynomial Division Algorithm

When we divide a polynomial $a(x)$ by a polynomial $b(x)$ , we get:

$a(x) = b(x) \times Q(x) + R(x)$

where:

$a(x)$ is the dividend polynomial
$b(x)$ is the divisor polynomial $(\neq 0)$
$Q(x)$ is the quotient polynomial
$R(x)$ is the remainder polynomial

In words: the dividend equals the divisor multiplied by the quotient, plus the remainder.

What are cubic polynomials?

A cubic polynomial is a polynomial expression where the highest power of the variable is 3. This means the degree of the polynomial is 3.

General form

The general form of a cubic polynomial is:

$y = ax^3 + bx^2 + cx + d$

where $a \neq 0$ (if $a = 0$ , it wouldn't be cubic anymore).

The term with $x^3$ is called the leading term, and ' $a$ ' is the leading coefficient.

Connection to previous learning

You may remember from Grade 10 how to factorise the sum and difference of cubes. These patterns become very useful when working with cubic polynomials.

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Sum and Difference of Cubes (Grade 10 Review)

Sum of cubes: $p^3 + 8 = (p + 2)(p^2 - 2p + 4)$
Difference of cubes: $k^3 - 1 = (k - 1)(k^2 + k + 1)$

These patterns are useful when working with cubic polynomials.

Methods for factoring cubic polynomials

There are three main methods we use to factorise cubic polynomials with one variable:

Long division
Synthetic division
Inspection

We'll focus on the first two methods, as they provide systematic approaches to polynomial division.

Long division method

Long division with polynomials works similarly to long division with numbers. It's a step-by-step process that finds both the quotient and remainder systematically.

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Worked Example: Long Division

Question: Use long division to find the quotient $Q(x)$ and remainder $R(x)$ when $a(x) = 2x^3 - x^2 - 6x + 16$ is divided by $b(x) = x - 1$ .

Solution:

Step 1: Write the division in standard form

We want: $a(x) = b(x) \times Q(x) + R(x)$

So: $2x^3 - x^2 - 6x + 16 = (x - 1) \times Q(x) + R(x)$

Step 2: Perform the long division

Make sure both polynomials are written in descending order of exponents. If any degree is missing, write it with coefficient 0.

         $2x^2 + x - 5$
$x - 1$ | $2x^3 - x^2 - 6x + 16$
         $2x^3 - 2x^2$
        ___________
            $x^2 - 6x$
            $x^2 - x$
           _______
               $-5x + 16$
               $-5x + 5$
              _______
                  $11$

Step 3: Write the final answer

$Q(x) = 2x^2 + x - 5$
$R(x) = 11$
Therefore: $a(x) = (x - 1)(2x^2 + x - 5) + 11$

Synthetic division method

Synthetic division is a simpler and more efficient method for dividing polynomials. It allows us to find the quotient and remainder by working with coefficients only, without needing to write out variables and exponents.

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Key Features of Synthetic Division

The coefficients of the dividend are written below a horizontal line
The coefficients of the quotient are written above the horizontal line
We add coefficients instead of subtracting (unlike long division)
We use the opposite sign of the constant term in the divisor
The coefficient of $x$ in the divisor must be 1

Table

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Worked Example: Synthetic Division

Question: Use synthetic division to find $Q(x)$ and $R(x)$ when $a(x) = 2x^3 - x^2 - 6x + 16$ is divided by $b(x) = x - 1$ .

Solution:

Step 1: Set up the synthetic division

Since we're dividing by $(x - 1)$ , we use $+1$ in our calculation.

$1 \;|\; 2 \;\; -1 \;\; -6 \;\; 16$

Step 2: Perform the synthetic division

$1 \;|\; 2 \;\;\; -1 \;\;\; -6 \;\;\; 16$
↓ + $2$ + $1$ - $5$
$\,\;\;\;2 \;\;\;\;\; +1 \;\;\;\;\; -5 \;\;\;\;\; 11$

Working through:

$q_2 = 2$
$q_1 = -1 + (2)(1) = 1$
$q_0 = -6 + (1)(1) = -5$
$R = 16 + (-5)(1) = 11$

Step 3: Write the final answer Since we divided a cubic by a linear expression, the quotient is quadratic:

$Q(x) = 2x^2 + x - 5$
$R(x) = 11$
Therefore: $a(x) = (x - 1)(2x^2 + x - 5) + 11$

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Another Synthetic Division Example

Question: Find $Q(x)$ and $R(x)$ when $a(x) = 6x^3 + x^2 - 4x + 5$ is divided by $b(x) = 2x - 1$ .

Solution:

Step 1: Prepare for synthetic division

First, we need the leading coefficient of the divisor to be 1:

$b(x) = 2x - 1 = 2(x - \frac{1}{2})$

So we divide by $(x - \frac{1}{2})$ and use $+\frac{1}{2}$ .

Step 2: Perform synthetic division

$\tfrac{1}{2} \;|\;\; 6 \;\;\; 1 \;\;\; -4 \;\;\; 5$
↓ $3$ $2$ $-1$
$\;\;\;\; 6 \;\;\; 4 \;\;\; -2 \;\;\; 4$

Step 3: Account for the factor of 2 Since our original divisor was $2(x - \frac{1}{2})$ , our final answer is:

$Q(x) = 6x^2 + 4x - 2 = 2(3x^2 + 2x - 1)$
$R(x) = 4$
Therefore: $a(x) = (2x - 1)(3x^2 + 2x - 1) + 4$

Choosing between methods

Both long division and synthetic division give the same results, but synthetic division is generally faster and less prone to arithmetic errors.

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When to Use Each Method

Synthetic division only works when:

The divisor is linear (degree 1)
The coefficient of $x$ in the divisor is 1 (or can be made 1 by factoring)

For more complex divisors, long division is necessary.

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Key Points to Remember:

A cubic polynomial has degree 3, with general form $y = ax^3 + bx^2 + cx + d$ where $a \neq 0$
The division algorithm for polynomials states that $a(x) = b(x) \times Q(x) + R(x)$
Long division works for any polynomial division but can be lengthy
Synthetic division is faster but only works for linear divisors with leading coefficient 1
Both methods will give you the same quotient and remainder when applied correctly

Cubic Polynomials (Grade 12 NSC Matric Mathematics): Revision Notes