What is a polynomial?
A polynomial in x is a sum of terms like 3x², −5x, 7, where the powers of x are whole numbers (0, 1, 2, …). The highest power is the degree.
- Degree 1: linear, ax + b (a ≠ 0)
- Degree 2: quadratic, ax² + bx + c (a ≠ 0)
- Degree 3: cubic, ax³ + bx² + cx + d
x + 1/x and √x + 2 are not polynomials (powers −1 and ½).
Zero of a polynomial
A number k is a zero of p(x) if p(k) = 0. For p(x) = x² − 2x − 3: p(3) = 9 − 6 − 3 = 0, so 3 is a zero. For a linear ax + b the zero is −b/a.
Geometric meaning of zeroes
Draw y = p(x). The zeroes are the x-coordinates of the points where the graph meets the x-axis, because there y = 0.
- Linear: a straight line, exactly one zero.
- Quadratic: a parabola (opens up if a > 0, down if a < 0). It can cut the x-axis at 2 points, touch it at 1 point, or miss it (0 zeroes).
- Cubic: at most 3 zeroes (it meets the axis at least once).
Rule: a polynomial of degree n has at most n zeroes. To count zeroes from a graph, count the points on the x-axis.
Relationship between zeroes and coefficients
If α and β are zeroes of ax² + bx + c, then ax² + bx + c = a(x − α)(x − β) = a[x² − (α+β)x + αβ]. Comparing terms:
α + β = −b/a and αβ = c/a
Check with x² − 2x − 3 = (x + 1)(x − 3): zeroes −1, 3. Sum 2 = −(−2)/1 ✓. Product −3 = −3/1 ✓.
(Extra: for a cubic ax³ + bx² + cx + d, α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a.)
Finding a quadratic from its zeroes
If the sum S and product P of zeroes are known, a quadratic is k[x² − Sx + P] for any non-zero k. Example: S = 4, P = 1 gives x² − 4x + 1.
Try it: a practical
Predict first, then check in the 3D free play: set a = 1, b = −5, c = 6. Where will the red balls sit? (Hint: which two numbers add to 5 and multiply to 6?) Then slowly raise c and note the value where the two balls meet. At home, throw a ball gently and sketch its path: the two ground points are like zeroes.
Board exam corner
Polynomials is part of Algebra (20 marks). Typical questions: count zeroes from a given graph (1 mark), find zeroes of a quadratic and verify the relation with coefficients (3 marks), find a quadratic with given sum and product (2 marks), and find a value like α² + β² or 1/α + 1/β using the relations (3 marks).
Key formulas and definitions
- Zero: p(k) = 0
- Linear ax + b: zero = −b/a
- α + β = −b/a
- αβ = c/a
- Quadratic with zeroes α, β: k[x² − (α+β)x + αβ]
- α² + β² = (α+β)² − 2αβ
Worked examples
1. How many zeroes does the graph show if a parabola touches the x-axis at one point only?
One zero (the two zeroes are equal). The number of zeroes = number of points where the graph meets the x-axis.
2. Find the zeroes of x² + 7x + 10 and verify the relationship.
x² + 7x + 10 = (x + 2)(x + 5). Zeroes −2, −5. Sum −7 = −b/a = −7/1 ✓. Product 10 = c/a = 10 ✓.
3. Find the zeroes of 6x² − 3 − 7x and verify.
6x² − 7x − 3 = 6x² − 9x + 2x − 3 = 3x(2x − 3) + 1(2x − 3) = (3x + 1)(2x − 3). Zeroes −1/3, 3/2. Sum 7/6 = −(−7)/6 ✓. Product −1/2 = −3/6 ✓.
4. Find the zeroes of 4u² + 8u.
4u(u + 2) = 0, so u = 0 or −2. Sum −2 = −8/4 ✓. Product 0 = 0/4 ✓.
5. Find a quadratic polynomial whose zeroes have sum −1/4 and product 1/4.
k[x² + (1/4)x + 1/4]. Take k = 4: 4x² + x + 1.
6. If α, β are zeroes of x² − 5x + 6, find α² + β² and 1/α + 1/β.
α + β = 5, αβ = 6. α² + β² = 25 − 12 = 13. 1/α + 1/β = (α+β)/αβ = 5/6.
7. If one zero of x² − 6x + k is twice the other, find k.
Zeroes α, 2α. Sum 3α = 6 ⇒ α = 2. Product 2α² = 8 = k. So k = 8.
Common mistakes
- Writing α + β = b/a. The sum has a minus sign: −b/a.
- Not arranging terms first: in 6x² − 3 − 7x, b is −7 and c is −3.
- Counting where the graph cuts the y-axis instead of the x-axis.
- Thinking every quadratic has two real zeroes. Some touch the axis (1) or miss it (0).