Quadratics and Polynomials Flashcards
What is the general form of a quadratic equation?
ax² + bx + c = 0, where a ≠ 0.
What is the quadratic formula?
x = [-b ± √(b² - 4ac)] / (2a)
What is the discriminant of a quadratic and what does it tell us?
b² - 4ac; it tells the nature of the roots: >0 = 2 real roots, =0 = 1 real root, <0 = complex roots.
What does it mean if a quadratic has no real roots?
The discriminant is less than zero.
What is completing the square used for?
Solving quadratics and identifying turning points of parabolas.
Complete the square: x² + 8x + 3
(x + 4)² - 13
What is the vertex of y = (x - h)² + k?
The point (h, k), which is the turning point of the parabola.
What is the axis of symmetry of a parabola?
A vertical line through the vertex, x = -b/(2a).
How do you solve a quadratic inequality?
Solve the corresponding equation, then test intervals on a number line.
What is a repeated root?
When a quadratic has only one real solution (discriminant = 0).
What is the factor theorem?
If f(a) = 0, then (x - a) is a factor of f(x).
What is the remainder theorem?
If a polynomial f(x) is divided by (x - a), the remainder is f(a).
How do you divide polynomials?
Use long division or synthetic division.
What is a cubic polynomial?
A polynomial of degree 3, typically with up to 3 real roots.
What is the general shape of a positive cubic graph?
Starts from the bottom left and rises to the top right.
What does it mean if a cubic has a repeated root?
One root is repeated; the graph touches the x-axis at that point.
How do you factorise a cubic polynomial?
Use the factor theorem to find one root, then factorise the quadratic remainder.
What is meant by a ‘turning point’ in a polynomial graph?
A local maximum or minimum where the graph changes direction.
What is the end behaviour of a polynomial?
Determined by the leading term; e.g., x³ grows to +∞ as x → ∞ if coefficient is positive.
What is a quartic polynomial?
A polynomial of degree 4, which can have up to 4 real roots.
