Polynomials

What is a Polynomial?

A polynomial is an algebraic expression made of one or more terms, where each term has a variable raised to a non-negative integer power multiplied by a coefficient.

General form: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

Examples: 4x³ - 2x² + 7, x + 5, 3x² - x, 6

Not polynomials: x⁻¹ (negative power), √x = x^(1/2) (fractional power), 1/x (same as x⁻¹)

Key Terms

Term	Meaning	Example
Degree	Highest power of the variable	4x³ - 2x² + 7 has degree 3
Coefficient	Number multiplying the variable	In 4x³, coefficient = 4
Constant term	Term with no variable	In 5x³ - 3x + 8, constant = 8
Leading coefficient	Coefficient of the highest-degree term	In -3x⁴ + 7x², leading = -3
Monomial	Polynomial with 1 term	4x³
Binomial	Polynomial with 2 terms	x + 5
Trinomial	Polynomial with 3 terms	3x² - x + 1

Adding Polynomials

To add polynomials, collect like terms — terms with the same variable raised to the same power.

(3x² + 2x - 1) + (x² - 5x + 4)

Group like terms:

• x² terms: 3x² + x² = 4x²
• x terms: 2x - 5x = -3x
• constants: -1 + 4 = 3

Result: 4x² - 3x + 3

Subtracting Polynomials

Distribute the negative sign to every term in the second bracket first, then collect like terms.

(2x² - x + 3) - (x² + 4x - 1)

= 2x² - x + 3 - x² - 4x + 1

• x² terms: 2x² - x² = x²
• x terms: -x - 4x = -5x
• constants: 3 + 1 = 4

Result: x² - 5x + 4

Multiplying Polynomials — FOIL

For two binomials, use FOIL (First, Outer, Inner, Last):

(x + 3)(x - 2):

• First: x × x = x²
• Outer: x × (-2) = -2x
• Inner: 3 × x = 3x
• Last: 3 × (-2) = -6

Combine: x² - 2x + 3x - 6 = x² + x - 6

Worked Example — Multiplying a Binomial by a Trinomial

(2x + 1)(x² - 3x + 4)

Multiply each term in the first bracket by each term in the second:

• 2x × x² = 2x³
• 2x × (-3x) = -6x²
• 2x × 4 = 8x
• 1 × x² = x²
• 1 × (-3x) = -3x
• 1 × 4 = 4

Collect like terms: 2x³ + (-6x² + x²) + (8x - 3x) + 4 = 2x³ - 5x² + 5x + 4

Special Products

These identities appear so frequently they are worth memorising:

• (a + b)² = a² + 2ab + b² (perfect square, sum)
• (a - b)² = a² - 2ab + b² (perfect square, difference)
• (a + b)(a - b) = a² - b² (difference of two squares)

Example using difference of squares: (x + 5)(x - 5) = x² - 25

This avoids doing full FOIL when you recognise the pattern.

Evaluating a Polynomial

To evaluate a polynomial at a given value, substitute the value in for x.

Find the value of 2x² - 3x + 1 when x = 4: = 2(4²) - 3(4) + 1 = 2(16) - 12 + 1 = 32 - 12 + 1 = 21

Common Mistakes to Avoid

• Forgetting to distribute the negative sign: when subtracting polynomials, every term in the subtracted bracket changes sign
• Adding unlike terms: x² and x are NOT like terms — you cannot combine them into x³
• Misapplying (a + b)² = a² + b²: the middle term 2ab is always missing when students forget to expand properly
• Wrong degree: the degree of a polynomial is the highest power after collecting like terms

Tips and Tricks

• Write terms in descending order (highest power first) to make comparing and adding easier
• Box or underline like terms with the same marking before combining
• Use FOIL as a checklist — tick off each of the four multiplications to avoid missing one
• Remember: (a + b)² always has three terms, never two