Given: $$f(x) = x^3 + 4x^2 - 7x - 10$$ 8.1 Write down the y-intercept of $f$ - NSC Mathematics - Question 8 - 2023 - Paper 1

To confirm that $2$ is a root, we compute:

$f(2) = (2)^3 + 4(2)^2 - 7(2) - 10 = 8 + 16 - 14 - 10 = 0.$
Since $f(2) = 0$, $2$ is indeed a root of the equation.

We already established that $(x - 2)$ is a root. We can use polynomial long division to divide $f(x)$ by $(x - 2)$:

$f(x) = (x - 2)(x^2 + 6x + 5).$
Now, we factor $x^2 + 6x + 5$:

$x^2 + 6x + 5 = (x + 1)(x + 5).$
Thus, the complete factorization is:

$f(x) = (x - 2)(x + 1)(x + 5).$

For the sketch of the graph, plot the y-intercept $(0, -10)$, the turning points $(0.7, 12.6)$ and $(-3.4, 20.8)$, and the x-intercepts found from the factorization: $(2, 0)$, $(-1, 0)$, and $(-5, 0)$.
Connect these points smoothly to illustrate the shape of the cubic function, ensuring to maintain the characteristics of turning points.

From the graph analysis, identify the sections where the curve is decreasing. This occurs when x is between the turning points, specifically for intervals approximately $(-3.4, 0.7)$.

To find where the gradient is at a minimum point, calculate the derivative $f'(x)$ and analyze critical points.
The minimum gradient is reached at the turning point $(-3.4, 20.8)$, indicating that the gradient transitions from negative to positive.