Back to: Further Mathematics SS2

**Graphs of cubic equations**

Polynomials of degree three have the general form y = ax^{3} + bx^{2} + cx + d(a ≠ 0). The curve is usually called a **cubical parabola**.

A cubical parabola has two shapes depending on whether a > 0 or a < 0.

Sketch each of the following curves represented by the following functions:

(a) y = x^{3} + 2x^{2} – 5x – 6

(b) y = 12 + 4x – 3x^{2} – x^{3}

**Solution**

(a) y = x^{3} + 2x^{2} – 5x – 6

Using the factor theorem and long division the expression can be factorized as:

y = (x – 2)(x + 1)(x + 3)

The zeros of the polynomial are therefore x = 2, x = -1 and x = 3, hence the x – intercepts are (2, 0), (-1, 0) and (-3, 0).

The y-intercept is (0, -6).

Next, we shall consider the behavior of the function at different intervals along the x-axis. This will enable us to see whether the curve is above or below the axis.

Mark the x-intercepts on the x-axis.

The intervals we shall consider are:

(a) x< -3

(b) -3 < x < -1

(c) -1 < x < 2

(d) x> 2

We shall examine the signs (+ve or –ve) in each of the intervals.

x = -4 is in the intervals x < -3

f(-4) = -18 < 0

Hence the part of the graph in the interval x < -3 is below the axis.

x = -2 is in the interval -3 < x <-1

f(-2) = 4 > 0

Hence the part of the graph in the interval -3 < x < 2 is above the x – axis x = 0 is in the interval -1 < x < 2

f(0) = -6 < 0.

Hence the part of the graph in the interval -1 < x < 2 is below the x – axis.

x = 3 is in the interval x > 2

f(3) = 24 > 0

Hence the part of the graph in the interval is above the x – axis.

The intercept on the axis coupled with the behaviour of the function at different intervals on the x – axis will enable us to get the shape of the curve.

The intervals we shall consider are:

(a) x< -3

(b) -3 < x < -1

(c) -1 < x < 2

(d) x> 2

x = -4 is in the interval x < -3

f(-4) = 12 < 0,

hence the part of the graph within this interval is above the x – axis.

x = -2.5 is in the interval -3 < x < -2

f(-2.5) =

hence the part of the graph within this interval -3 < x < -2 is above the axis.

x = 0 is in the interval -2 < x < 2

f(0) = 12 > 0,

hence the part of the graph within this interval -2< x <2 is above the axis.

x = 3 is in the interval x > 2

f(3) = -30< 0,

hence the part of the graph within the interval r > 2 is below the x – axis.

**Weekend Assignment**

Given that a cubic equation x^{3} + 2x^{2} – 19x – 20 = 0 has 4 as one its roots, find the

(1) Second root (a) -1 (b) 1 (c) 2 (d) 3

(2) Third root (a) 5 (b) -8 (c) 3 (d) 2

(3) Sum of the second and third roots (a) -4 (b) 6 (c) 4 (d) -6

(4) Product of the second and third roots (a) 6 (b) 5 (c) -6 (d) -5

(5) Find the zeros of x^{2} – 1 (a) 2 or -2 (b) 1 or 2 (c) -1 or 1 (d) 1 or -2

**Theory**

(1) If (x + 1) is a factor of f(x) = x^{3} + kx^{2} + 3x + 10, find the value of the constant k.

(2) Factorise f(x) completely.

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