Algebra GCSE level 9 - Sketching cubic graph

A level 9 algebra GCSE question.

Solve the cubic equation 2x^3 + 5x^2 - x – 6 = 0, given that x = 1 is one solution
Hence sketch the curve y = 2x^3 + 5x^2 - x – 6

The meaning of sketch a cubic equation.

Making a sketch means that it is not drawn to scale, the axes are not numerically labelled. Only the points where the cubic equation intersects with the x and y axes are shown and labelled.

Using a quadratic equation to solve the cubic equation.

Using the information given in this algebra gcse question that x = 1 is one solution, the video shows that (x - 1) must be a factor.
The highest term in the factor is x, the highest term in a quadratic is x^2, multiplying the two highest terms gives x^3. So (x - 1) multiplied by a quadratic gives a cubic equation.
Since we don’t know what the quadratic is we use a basic formula for a quadratic; ax^2 + bx + c, where a, b and c are constants.

Finding the values of the constants in the quadratic equation formula.

The multiplication of the factor (x – 1) and the quadratic equation is clearly shown in the video.
The resulting cubic equation involving the constants a, b and c is then compared with the cubic equation in the question. Equating the coefficients provides simple linear equations which are solved, as explained in the video. The values are then substituted in the basic quadratic equation.

Solving the quadratic equation by factorising.

The video clearly shows the factorisation of the quadratic. A total of three factors of the cubic equation are eventually obtained.

Finding the points of intersection on the x-axis where y = 0.

Equating each factor to zero, since this implies when y = 0, gives simple linear expressions which are easily solved, as explained in the video.

Finding the points of intersection on the y-axis where x = 0.

It is necessary to find if and where the cubic curve cuts the y-axis, calculated by substituting x = 0 into the cubic equation y = 2x^3 + 5x^2 - x – 6. Since all the terms involving x will now be zero, the value of y must equal -6.

Plotting the points on the axes and using to sketch the cubic equation.

The video shows all the points of intersection plotted on the axes, which are subsequently joined with a smooth curve to produce a sketch of the cubic equation.