A quick recap
A polynomial in x has terms of the form axⁿ with n a whole number. Its degree is the highest power with a non-zero coefficient.
Linear: ax + b
Quadratic: ax² + bx + c (a ≠ 0)
Cubic: ax³ + bx² + cx + d (a ≠ 0)
A zero of p(x) is a number k with p(k) = 0.
The geometric meaning of zeroes
Plot y = p(x). The zeroes of p(x) are exactly the x-coordinates of the points where the graph meets the x-axis, because y = 0 there.
A linear polynomial ax + b has a straight-line graph that crosses the x-axis once, at x = −b/a. It has exactly one zero.
A quadratic ax² + bx + c has a U-shaped graph called a parabola. It opens upward if a > 0 and downward if a < 0. It can meet the x-axis at two points, touch it at one point, or miss it completely, so a quadratic has at most two zeroes.
A cubic has at most three zeroes.
In general, a polynomial of degree n has at most n zeroes.
Zeroes and coefficients of a quadratic
If α and β are the zeroes of ax² + bx + c, then
α + β = −b/a and αβ = c/a
Example. Find the zeroes of x² − 2x − 8 and check the relations.
x² − 2x − 8 = (x − 4)(x + 2), so the zeroes are 4 and −2.
Sum = 4 + (−2) = 2, and −b/a = −(−2)/1 = 2. ✓
Product = 4 × (−2) = −8, and c/a = −8/1 = −8. ✓
Example. Find the zeroes of 6x² − 7x − 3.
Split the middle term: we need two numbers with product 6 × (−3) = −18 and sum −7. They are −9 and 2.
6x² − 9x + 2x − 3 = 3x(2x − 3) + 1(2x − 3) = (2x − 3)(3x + 1). Zeroes: 3/2 and −1/3.
Sum = 3/2 − 1/3 = 7/6 = −b/a ✓, product = −1/2 = c/a ✓.
Forming a quadratic from its zeroes
A quadratic whose zeroes have sum S and product P is
k(x² − Sx + P), for any non-zero constant k.
Example. Find a quadratic whose zeroes have sum −3 and product 2.
x² − (−3)x + 2 = x² + 3x + 2. Check: x² + 3x + 2 = (x + 1)(x + 2), zeroes −1 and −2, sum −3, product 2. ✓
Example. Zeroes 1/4 and −1. Sum = −3/4, product = −1/4. The polynomial is x² + ¾x − ¼; multiplying by 4 gives 4x² + 3x − 1.
Using the relations without finding zeroes
If α and β are the zeroes of x² − 5x + 6, find α² + β².
α + β = 5 and αβ = 6. Since (α + β)² = α² + β² + 2αβ, we get α² + β² = 25 − 12 = 13.
Similarly, 1/α + 1/β = (α + β)/αβ = 5/6.
Cubic polynomials
If α, β and γ are the zeroes of ax³ + bx² + cx + d, then
α + β + γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a
Example. The zeroes of x³ − 6x² + 11x − 6 are 1, 2 and 3. Check: sum = 6 = −(−6)/1, sum of products in pairs = 2 + 6 + 3 = 11, product = 6 = −(−6)/1. ✓
Summary
Zeroes are where the graph of y = p(x) meets the x-axis; degree n gives at most n zeroes.
For ax² + bx + c: α + β = −b/a and αβ = c/a.
A quadratic with zeroes summing to S and multiplying to P is k(x² − Sx + P).
For a cubic, the sum of zeroes is −b/a and their product is −d/a.