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Polynomials

Degree and Zeroes of a Polynomial

What is a Polynomial?

A polynomial is an algebraic expression that contains one or more terms, where each term consists of a variable raised to a non-negative integer exponent, multiplied by a coefficient (a real number).

The word "polynomial" comes from "poly" (many) and "nomial" (terms).

Key Components of a Polynomial:

ComponentDefinitionExample in 3x² + 2x - 5
**Coefficient**The numerical factor of a term3, 2, -5
**Variable**The letter representing an unknown quantityx
**Exponent**The power to which the variable is raised2, 1, 0
**Degree**The highest exponent in the polynomial2

Polynomials in One Variable:

A polynomial in one variable (say x) has the general form:

aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ, aₙ₋₁, ..., a₀ are constants (coefficients).

Types of Polynomials Based on Degree:

TypeDegreeStandard FormExample
**Linear**1ax + b3x + 4, -2x + 1
**Quadratic**2ax² + bx + cx² - 5x + 6, 2x² + 3x - 1
**Cubic**3ax³ + bx² + cx + dx³ - 8, 2x³ + x² - x + 4

Types of Polynomials Based on Number of Terms:

TypeNumber of TermsExample
**Binomial**2x + 3, 2x² - 5, 4x³ + 1
**Trinomial**3x² + 5x + 6, 3x³ - 2x² + x
Anatomy of a Polynomial3x⁴ − 5x² + 2x − 7Leadingcoefficient: 3Degree (highestpower): 4Coefficientof x: 2Constantterm: -7Monomial1 term3x²Binomial2 termsx+5Trinomial3 termsx²+x+1Polynomialn terms
Example 1: State the degree of 4x³ − 7x + 2.
The highest power is 3, so the degree is 3 (cubic).
Example 2: Is x = 2 a zero of p(x) = x² − 5x + 6?
p(2) = 4 − 10 + 6 = 0, so yes, 2 is a zero.
Quick recap
  • Degree = highest power of the variable.
  • A zero of p(x) makes p(x) = 0; evaluate by substituting the value.
✓ Quick check
If a + b + c = 9 and ab + bc + ca = 26, then the value of a² + b² + c² is:
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca), so 81 = a² + b² + c² + 52, giving a² + b² + c² = 29.
What is the zero of the linear polynomial p(x) = cx + d, where c ≠ 0?
Set cx + d = 0, so cx = −d and x = −d/c.

Remainder and Factor Theorems

What is Polynomial Division?

Polynomial division is similar to long division with numbers. When dividing a polynomial by a linear divisor (x - a), we can use two methods:

  1. Long division (general method)
  2. Synthetic division (shortcut when divisor is x - a)

Division Algorithm for Polynomials:

When dividing polynomial P(x) by divisor d(x), we get:

P(x) = d(x) × Q(x) + R(x)

Where Q(x) is the quotient and R(x) is the remainder. The degree of R(x) is less than the degree of d(x).

What is a Zero of a Polynomial?

A zero (or root) of a polynomial P(x) is a value of x for which P(x) = 0.

  • If P(a) = 0, then a is a zero of the polynomial
  • Geometrically, a zero means the graph of the polynomial crosses the x-axis at x = a

Remainder Theorem:

When a polynomial P(x) is divided by (x - a), the remainder is P(a).

This means: P(x) = (x - a) × Q(x) + P(a)

Factor Theorem (special case of Remainder Theorem):

If P(a) = 0, then (x - a) is a factor of P(x).
Remainder TheoremRemainder Theorem: When p(x) is divided by (x - a),the remainder is p(a)Example: p(x) = x³ - 3x + 5, divide by (x - 2)Remainder Theorem: remainder = p(2)p(2) = (2)³ - 3(2) + 5 = 8 - 6 + 5 = 7∴ Remainder = 7Verify by Long Division:x³-3x+5 ÷ (x-2)Quotient: x²+2x+1Remainder: 7 ✓Applications: Test divisibility, find unknown coefficientsIf p(a) = 0, then (x - a) is a FACTOR of p(x)This is the Factor Theorem!
Example 1: Find the remainder when x³ + 1 is divided by (x + 1).
Here a = −1: p(−1) = (−1)³ + 1 = 0, so the remainder is 0.
Example 2: Is (x − 1) a factor of x³ − 1?
p(1) = 1 − 1 = 0, so by the Factor Theorem (x − 1) is a factor.
Quick recap
  • Remainder on dividing p(x) by (x − a) is p(a).
  • (x − a) is a factor ⟺ p(a) = 0.
✓ Quick check
If (x − 2) is a factor of x² + kx + 2k, then the value of k is:
By the Factor Theorem p(2) = 0: 4 + 2k + 2k = 0 → 4 + 4k = 0 → k = −1.
The expansion of (x − 2y − 3z)² is:
With a = x, b = −2y, c = −3z: squares x² + 4y² + 9z²; cross terms 2(x)(−2y) = −4xy, 2(−2y)(−3z) = +12yz, 2(−3z)(x) = −6zx.

Factorisation and Algebraic Identities

What is Factorization?

Factorization (or factoring) is the process of writing a polynomial as a product of simpler polynomials (factors). It's the reverse of multiplication.

Key Algebraic Identities for Factorization:

IdentityFormulaExample
**Perfect Square (Difference)**(x - y)² = x² - 2xy + y²(x - 5)² = x² - 10x + 25
**Difference of Squares**x² - y² = (x + y)(x - y)x² - 16 = (x + 4)(x - 4)
**Product of Binomials**(x + a)(x + b) = x² + (a + b)x + ab(x + 3)(x + 4) = x² + 7x + 12

Steps to Factor a Quadratic Trinomial (x² + bx + c):

  1. Find two numbers whose product = c and sum = b
  2. Write as (x + m)(x + n) where m × n = c and m + n = b

Special Cases:

  • If a ≠ 1 in ax² + bx + c, factoring requires more steps (splitting middle term)
  • Difference of squares: a² - b² = (a - b)(a + b)
  • Sum of squares a² + b² cannot be factored with real numbers
Factor Theorem & FactorisationFactor Theorem: (x - a) is a factor of p(x) ⟺ p(a) = 0i.e. a is a zero (root) of the polynomialFactorise: p(x) = x³ - 6x² + 11x - 6Try x=1: p(1) = 1 - 6 + 11 - 6 = 0 ✓ → (x-1) is a factorDivide: x³-6x²+11x-6 ÷ (x-1) = x²-5x+6Factorise x²-5x+6 = (x-2)(x-3)∴ p(x) = (x-1)(x-2)(x-3)Zeros of the polynomial: x = 1, 2, 3x=1x=2x=3
Example 1: Factorise x² + 7x + 12.
Split 7 as 3 + 4: x² + 3x + 4x + 12 = (x + 3)(x + 4).
Example 2: Using an identity, expand (a + b + c)² when a = b = c = 1.
1 + 1 + 1 + 2(1 + 1 + 1) = 3 + 6 = 9, which is (1+1+1)² = 9.
Quick recap
  • Split the middle term to factorise quadratics.
  • Know (x+y+z)² and the x³+y³+z³−3xyz identity.
✓ Quick check
If x = −1 is a zero of p(x) = kx³ − 2x² + x + 4, find k.
p(−1) = −k − 2 − 1 + 4 = −k + 1 = 0, so k = 1.
Find the remainder when p(x) = 2x³ − 7x² + 9x − 2 is divided by (2x − 1).
By the Remainder Theorem put x = 1/2: p(1/2) = 2(1/8) − 7(1/4) + 9(1/2) − 2 = 1/4 − 7/4 + 18/4 − 8/4 = 4/4 = 1.
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