Revision Revision Notes for Grade 12 NSC Matric Mathematics

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What are polynomials?

A polynomial is a mathematical expression containing one or more variables with different powers, where each variable has a coefficient. Understanding polynomials is crucial for success in NSC Mathematics Paper 1.

The general form of a polynomial is: $a\_n \times x^n + a\_{n-1} \times x^{n-1} + ... + a\_2 \times x^2 + a\_1 \times x + a\_0$

Where $n$ must be a natural number (including zero).

Key polynomial terminology

Understanding the basic terminology is essential for working with polynomials effectively.

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Key Terms:

Variables are symbols representing unknown numerical values (like $a$ , $b$ , $x$ , $y$ , $\alpha$ , $\theta$ )
Coefficients are the numbers multiplied by the variables in an expression
Expressions are combinations of numbers, variables and basic operators (+, −, ×, ÷)
Univariate expressions contain only one variable

Types of polynomials

Polynomials are classified by the number of terms they contain. This classification helps us understand the structure and complexity of different polynomial expressions.

Monomial

A polynomial with one term only. Examples include:

$7a^2b$
$15xyz^2$
$-3x^4$

Binomial

A polynomial with two terms. Examples include:

$2x + 5z$
$26 - y^2k$
$x^2 - 9$

Trinomial

A polynomial with three terms. Examples include:

$a - b + c$
$4x^2 + 17xy - y^3$
$2x^2 + 5x - 3$

Degree or order

The degree (also called order) of a polynomial is the highest exponent of the variable. For example:

$7p - 12p^2 + 3p^5 + 8$ has degree 5
$2x^3 - 4x + 1$ has degree 3

Identifying polynomials vs non-polynomials

Being able to distinguish between polynomials and non-polynomials is a fundamental skill that appears frequently in exams.

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Critical Rule: All exponents in a polynomial must be natural numbers (0, 1, 2, 3, 4, ...).

If you see negative exponents, fractional exponents, or variables in denominators, the expression is not a polynomial.

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These are NOT polynomials:

$\frac{3}{y} - 4y^2 + 1$ (negative exponent: $y^{-1}$ )
$5\sqrt{k} + k - 2k^2$ (fractional exponent: $k^{1/2}$ )
$x^{-2} + 3 + 7x^2$ (negative exponent)
$t^2 - 4t + 6t^{1/4}$ (fractional exponent)

Quadratic polynomials

Quadratic polynomials have degree 2 and are extremely important for NSC exams. These polynomials appear in many contexts and mastering their manipulation is essential for success.

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General factorisation techniques:

Taking out common factors
Factorising difference of squares: $a^2 - b^2 = (a + b)(a - b)$
Grouping in pairs
Sum and difference of cubes

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Quadratic-specific methods:

Factorisation by inspection (finding factors that multiply and add correctly)
Completing the square
Quadratic formula
Suitable substitution

These methods are essential for solving quadratic equations and will be used extensively in later chapters.

Solving quadratic equations

Mastering different solution methods gives you flexibility in choosing the most efficient approach for each problem.

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Worked Example 1: Solving by factorisation

Question: Solve for $x$ : $\frac{3x}{x + 2} + 1 = \frac{4}{x + 1}$

Solution:

Step 1: Determine restrictions

The denominators cannot equal zero, so $x \neq -2$ and $x \neq -1$ .

Step 2: Find the lowest common denominator

LCD = $(x + 2)(x + 1)$

Step 3: Multiply through and simplify

$3x(x + 1) + (x + 2)(x + 1) = 4(x + 2)$
$3x^2 + 3x + x^2 + 3x + 2 = 4x + 8$
$4x^2 + 2x - 6 = 0$
$2x^2 + x - 3 = 0$

Step 4: Factorise and solve

$(2x + 3)(x - 1) = 0$
Therefore: $x = -\frac{3}{2}$ or $x = 1$

Step 5: Check restrictions and write final answer

Both solutions are valid.
Therefore: $x = -1\frac{1}{2}$ or $x = 1$

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Worked Example 2: Using the quadratic formula

Question: Find the roots of $f(x) = 3x^2 + 4x - 8$

Solution:

Step 1: Set equal to zero

$3x^2 + 4x - 8 = 0$

Step 2: Check if factorisable

Cannot be easily factored, so use the quadratic formula.

Step 3: Identify coefficients

$a = 3$ , $b = 4$ , $c = -8$

Step 4: Apply the quadratic formula

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

$x = \frac{-4 \pm \sqrt{16 - 4(3)(-8)}}{2(3)}$

$x = \frac{-4 \pm \sqrt{16 + 96}}{6}$

$x = \frac{-4 \pm \sqrt{112}}{6}$

$x = \frac{-4 \pm 4\sqrt{7}}{6}$

$x = \frac{-2 \pm 2\sqrt{7}}{3}$

Step 5: Write final answer

$x = \frac{-2 + 2\sqrt{7}}{3}$ or $x = \frac{-2 - 2\sqrt{7}}{3}$

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Worked Example 3: Completing the square

Question: Solve by completing the square: $y^2 - 10y - 11 = 0$

Solution:

Step 1: Ensure coefficient of $y^2$ equals 1

Already satisfied: $y^2 - 10y - 11 = 0$

Step 2: Take half the coefficient of $y$ and square it

Coefficient of $y = -10$

Half of $-10 = -5$

$(-5)^2 = 25$

Step 3: Add and subtract this value

$y^2 - 10y + 25 - 25 - 11 = 0$

Step 4: Write as a perfect square

$(y - 5)^2 - 36 = 0$

$(y - 5)^2 = 36$

Step 5: Take square roots of both sides

$y - 5 = \pm\sqrt{36}$

$y - 5 = \pm 6$

Step 6: Solve for $y$

If $y - 5 = 6$ , then $y = 11$

If $y - 5 = -6$ , then $y = -1$

Final answer: $y = 11$ or $y = -1$

chatImportant

Important exam tip: Always remember that taking square roots gives both positive and negative answers.

bookmarkSummary

Key Points to Remember:

Polynomials must have natural number exponents only - no fractions, negatives, or variables in denominators
Degree equals the highest exponent in the polynomial
Mono = 1 term, Bi = 2 terms, Tri = 3 terms
Always check restrictions first when solving rational equations
The quadratic formula works for any quadratic equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Revision (Grade 12 NSC Matric Mathematics): Revision Notes

Revision

What are polynomials?

Key polynomial terminology

Types of polynomials

Monomial

Binomial

Trinomial

Degree or order

Identifying polynomials vs non-polynomials

Quadratic polynomials

Solving quadratic equations

Explore Grade 12 NSC Matric Mathematics Model Answers by Topics

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