What this chapter is about
This chapter extends your understanding of polynomials beyond what you studied in earlier classes. You learn to connect a polynomial's algebraic expression with its geometric picture — the graph. The focus is on how the zeroes of a polynomial relate to where its graph crosses the x-axis.
You study the relationship between the zeroes of quadratic and cubic polynomials and the coefficients appearing in those polynomials. This connection lets you find zeroes from coefficients or verify your answers without fully solving the polynomial.
By the end, you should be able to find zeroes of a quadratic polynomial by factorisation, understand what the graph of a linear, quadratic or cubic polynomial looks like, and use the sum-and-product relationships between zeroes and coefficients confidently.
Key ideas
- A polynomial in one variable x is an expression like 3x² − 5x + 2 where the powers of x are whole numbers (0, 1, 2, …) and coefficients are real numbers.
- The degree of a polynomial is the highest power of the variable with a non-zero coefficient: degree 1 is linear, degree 2 is quadratic, degree 3 is cubic.
- A zero (or root) of a polynomial p(x) is a value k such that p(k) = 0; geometrically, the graph of y = p(x) meets the x-axis at the point (k, 0).
- A linear polynomial ax + b (a ≠ 0) has exactly one zero, namely −b/a.
- A quadratic polynomial ax² + bx + c can have at most two zeroes; its graph is a parabola opening upward if a > 0 and downward if a < 0.
- A cubic polynomial can have at most three zeroes; its graph is a smooth curve that can cross the x-axis up to three times.
- The zeroes of a polynomial are found by factorising it or by using the relationship between zeroes and coefficients.
- Division algorithm for polynomials: for polynomials p(x) and g(x) with g(x) ≠ 0, there exist polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).
Formulas and facts to remember
For a quadratic polynomial ax² + bx + c with zeroes α and β:
- Sum of zeroes: α + β = −b/a (negative of coefficient of x divided by coefficient of x²)
- Product of zeroes: αβ = c/a (constant term divided by coefficient of x²)
For a cubic polynomial ax³ + bx² + cx + d with zeroes α, β and γ:
- Sum of zeroes: α + β + γ = −b/a
- Sum of products taken two at a time: αβ + βγ + γα = c/a
- Product of all three zeroes: αβγ = −d/a
A polynomial of degree n has at most n zeroes.
Worked examples
Example 1: Find the zeroes of the quadratic polynomial x² − 7x + 12 and verify the relationship between zeroes and coefficients.
Factorise: x² − 7x + 12 = (x − 3)(x − 4)
Setting each factor to zero: x − 3 = 0 gives x = 3; x − 4 = 0 gives x = 4.
So the zeroes are α = 3 and β = 4.
Verification: Here a = 1, b = −7, c = 12.
Sum of zeroes = 3 + 4 = 7. Also −b/a = −(−7)/1 = 7. ✓
Product of zeroes = 3 × 4 = 12. Also c/a = 12/1 = 12. ✓
Example 2: If the zeroes of 2x² − 5x + k are 2 and β, find β and the value of k.
Using sum of zeroes: 2 + β = −(−5)/2 = 5/2
So β = 5/2 − 2 = 5/2 − 4/2 = 1/2.
Using product of zeroes: 2 × (1/2) = k/2
So 1 = k/2, which gives k = 2.
Example 3: A rectangular plot has area 200 square metres. Its length is 10 metres more than its breadth. Form a polynomial whose zeroes give the dimensions.
Let breadth = x metres. Then length = (x + 10) metres.
Area = x(x + 10) = 200
So x² + 10x − 200 = 0.
Factorise: We need two numbers whose product is −200 and sum is 10. These are 20 and −10.
x² + 20x − 10x − 200 = x(x + 20) − 10(x + 20) = (x − 10)(x + 20)
Zeroes are x = 10 and x = −20. Since breadth cannot be negative, x = 10.
Breadth = 10 metres, length = 20 metres.
Common mistakes
Forgetting the negative sign in sum of zeroes: students write α + β = b/a instead of −b/a → always put the minus sign first, then substitute values.
Confusing zeroes with coefficients: a zero is a value of x that makes the polynomial zero, not the numbers appearing in the expression → substitute the zero into the polynomial and check you get 0.
Assuming every quadratic has two distinct real zeroes: some quadratics touch the x-axis once (repeated zero) or not at all → the graph shows whether zeroes exist and how many.
Writing the division algorithm incorrectly: students forget that the remainder must have degree strictly less than the divisor → after dividing, check that the remainder's degree is lower.
Sign errors when the leading coefficient is negative: for −x² + 4x − 3, students misread a, b, c → rewrite clearly: a = −1, b = 4, c = −3, then apply formulas.
Quick revision
- Degree tells the maximum number of zeroes a polynomial can have.
- For ax² + bx + c: sum of zeroes = −b/a, product of zeroes = c/a.
- Graph of a quadratic is a parabola; zeroes are where it crosses the x-axis.
- Division algorithm: dividend = divisor × quotient + remainder.
- Always verify zeroes by substituting back into the original polynomial.