Cubic expressions
Polynomial
= an algebraic expression with positive, while number exponents
The degree of a polynomial
= determines by the highest power of the variable
Linear polynomial
= polynomial in the 1st degree
= f(x) = ax + b
Quadratic polynomial
= polynomial in the 2nd degree
= f(x) = ax2 + bx + c
Cubic polynomial
= polynomial in the 3rd degree
- f(x) = ax3 + bx2 + cx + d
- Has three roots/zeros
- Always has one real and rational root
- The other two roots could be real, non real, rational, irrational, equal or unequal
- Can be factorized into a liner factor and a quadratic factor
Example: x3 - 1 = 0
(x - 1)(x2 + x + 1) = 0
Linear Quadratic
A non-zero constant
= a polynomial with a degree of zero
= f(x) = k where k ≠ 0
The zeros of a polynomial
= the values of x that will make f(x) = 0
Also knows as roots or x intercepts or solutions.
The Euclidean Property
Dividend = quotient x divisor + remainder
f(x) = b(x).q(x) + R(x)
The Remainder Theorem
−𝑏
If a polynomial f(x) is divided by a linear polynomial ax + b, the remainder R = f( 𝑎 )
The Factor Theorem
−𝑏
If f is a polynomial such that f( 𝑎 ) = 0, then ax + b is a factor of f(x)
−𝑏
Conversely, if ax + b is a factor of f(x), then f( 𝑎 ) = 0
,3 real, unequal roots
3 real roots, 2 equal
3 roots, all equal
1 real root, 2 non real
, Differential Calculus
The study of change
f(x) or y = x2 + 2x + 3
Is just a rule telling us what to do with x.
x = independent variable
y = dependant variable
f(x) = x2 + 2x + 3
f(x + h) = (x + h)2 + 2(x + h) + 3
f(2) = 22 + 2(2) + 3 = 11
This means that (2;11) is an ordered pair that lies on the graph and satisfies the equation.
Every point on this curve has the coordinates (a; f(a))
f(x+h)
(a : f(a))
f(x)
x x+h
Gradients:
Increasing if y increases as x increases.
An increasing line has a positive gradient.
Decreasing if y decreases as x increases.
An decreasing line has a negative gradient
Stationary
Increasing
Decreasing
Polynomial
= an algebraic expression with positive, while number exponents
The degree of a polynomial
= determines by the highest power of the variable
Linear polynomial
= polynomial in the 1st degree
= f(x) = ax + b
Quadratic polynomial
= polynomial in the 2nd degree
= f(x) = ax2 + bx + c
Cubic polynomial
= polynomial in the 3rd degree
- f(x) = ax3 + bx2 + cx + d
- Has three roots/zeros
- Always has one real and rational root
- The other two roots could be real, non real, rational, irrational, equal or unequal
- Can be factorized into a liner factor and a quadratic factor
Example: x3 - 1 = 0
(x - 1)(x2 + x + 1) = 0
Linear Quadratic
A non-zero constant
= a polynomial with a degree of zero
= f(x) = k where k ≠ 0
The zeros of a polynomial
= the values of x that will make f(x) = 0
Also knows as roots or x intercepts or solutions.
The Euclidean Property
Dividend = quotient x divisor + remainder
f(x) = b(x).q(x) + R(x)
The Remainder Theorem
−𝑏
If a polynomial f(x) is divided by a linear polynomial ax + b, the remainder R = f( 𝑎 )
The Factor Theorem
−𝑏
If f is a polynomial such that f( 𝑎 ) = 0, then ax + b is a factor of f(x)
−𝑏
Conversely, if ax + b is a factor of f(x), then f( 𝑎 ) = 0
,3 real, unequal roots
3 real roots, 2 equal
3 roots, all equal
1 real root, 2 non real
, Differential Calculus
The study of change
f(x) or y = x2 + 2x + 3
Is just a rule telling us what to do with x.
x = independent variable
y = dependant variable
f(x) = x2 + 2x + 3
f(x + h) = (x + h)2 + 2(x + h) + 3
f(2) = 22 + 2(2) + 3 = 11
This means that (2;11) is an ordered pair that lies on the graph and satisfies the equation.
Every point on this curve has the coordinates (a; f(a))
f(x+h)
(a : f(a))
f(x)
x x+h
Gradients:
Increasing if y increases as x increases.
An increasing line has a positive gradient.
Decreasing if y decreases as x increases.
An decreasing line has a negative gradient
Stationary
Increasing
Decreasing
