Cubic Polynomials

Rational Zero Theorem
If the rational number E/F, in lowest terms, is a zero of the polynomial 0111...)(a x a x a x a x P n n n n ++++=--, 0≠n a with integer coefficients, then E must be an integer factor of 0a and F must be an integer factor of n a .
1. Factorise 6116)(2
3-+-=x x x x f and hence solve the
equation f(x)=0.
Solution:
If E/F in lowest terms is a rational zero of f(x), then E must be a factor of -6 and F must be a factor of 1.
Possible values of E are the integer factors of -6: 6,3,2,1±±±±
Possible values of F are the integer factors of 1: 1±
Writing all possible fractions E/F, we have the possible rational zeros for f(x): 6,3,2,1±±±±
And using the factor theorem, we have f(1)=0 Therefore, (x-1) is a factor of f(x)
By synthetic division we obtain
)65)(1()(2+--=x x x x f Factorising the quadratic, we obtain
f(x)=(x-1)(x-2)(x-3)
For the equation f(x)=0
Therefore, x=1 or x=2 or x=3.
Cubic Polynomials
A cubic polynomial has form d cx bx ax y +++=23 where 0≠a and a, b, c and d are constants.
∙ If a > 0 the graph’s shape is_____________.
∙ If a < 0 the graph’s shape is_____________.
∙ For a cubic in the form ))()((γβα---=x x x a y the graph has x-intercepts γβα,,and the graph crosses over or cuts the x-axis at these points.
∙ For a cubic in the form
)()(2βα--=x x a y the graph touches the x-axis at αand cuts it at β.
∙ For a cubic in the form ))((2αγβ-++=x x x a y where the
discriminant of the quadratic factor is < 0, the graph cuts the x-axis once only at α
.。