Ib Math Hl Ia, Zeros of Cubic Functions

Mathematics IB Higher Level Portfolio (Internal Assessment):

Zeros of Cubic Functions

Brendan Lee

March 2010

Q1: Consider the Cubic equation:〖f(x)=2x〗^3+〖6x〗^2-4.5x-13.5:

The graph is shown here:

Graphed on GCalc 3.0

The zeros of this equation are:

x=-3

x=-1.5

x=1.5

This can be proved using the remainder theorem:

〖f(-3)=2(-3)〗^3+〖6(-3)〗^2-4.5(-3)-13.5

f(-3)=2(-27)+6(9)-(-13.5)—13.5

f(-3)=0

〖f(-1.5)=2(-1.5)〗^3+〖6(-1.5)〗^2-4.5(-1.5)-13.5

f(-1.5)=2(-3.375)+6(2.25)+6.75—13.5

f(-1.5)=0

〖f(1.5)=2(1.5)〗^3+〖6(1.5)〗^2-4.5(1.5)-13.5

f(1.5)=2(3.375)+6(2.25)-6.75—13.5

f(1.5)=0

Equations of tangent lines at average of two roots:

Roots -1.5 and -3:

Average: (-1.5+(-3))/2=-2.25

〖y=2(-2.25)〗^3+〖6(-2.25)〗^2-4.5(-2.25)-13.5

y≈-4.2

Slope at (-2.25, -4.2)

f^' (x)=〖6x〗^2+12x-4.5

m=〖6(-2.25)〗^2+12(-2.25)-13.5=-1.125

y – y1 = m(x – x1)

y –(-4.2)= -1.125(x –(-2.25))

Equation of tangent line: y=-1.125x+1.7

Graphed on GCalc 3.0

Roots 1.5 and -1.5:

Average: (-1.5+1.5)/2=0

〖y=2(0)〗^3+〖6(0)〗^2-4.5(0)-13.5

y=-13.5

Slope at (0, -13.5)

f^' (x)=〖6x〗^2+12x-4.5

m=〖6(0)〗^2+12(0)-4.5=4.5