Cubic equations questions with answers

Questions

1. Question 1Given a factor

   Show that $x - 3$ is a factor of $x^3 + x^2 - 9x - 9$, and hence solve $x^3 + x^2 - 9x - 9 = 0$.

   Show the worked solution

   1. Step 1: Factor theorem: (x - 3) is a factor if f(3) = 0$f(3) = (3)^3 + (3)^2 - 9(3) - 9$$= 27 + 9 - 27 - 9$$= 0$Since $f(3) = 0$, $(x - 3)$ is a factor of $f(x)$.
   2. Step 2: Write f(x) = (x - 3)(x² + px + q), expand the right-hand side and collect like terms$x^3 + x^2 - 9x - 9 = (x - 3)(x^2 + px + q)$$(x - 3)(x^2 + px + q) = x^3 + px^2 + qx - 3x^2 - 3px - 3q$$= x^3 + (p - 3)x^2 + (q - 3p)x - 3q$
   3. Step 3: Compare the coefficients of x², then the numbers$p - 3 = 1$$p = 4$$-3q = -9$$q = 3$Check the x term: $q - 3p = 3 - 3 \times 4$$q - 3p = -9$ ✓
   4. Step 4: Factorise the quadratic factor, then set each factor to zero$x^3 + x^2 - 9x - 9 = (x - 3)(x + 3)(x + 1)$$x = 3$$x = -3$$x = -1$
2. Question 2Find a factor first

   Solve $x^3 - 6x^2 + 11x - 6 = 0$.

   Show the worked solution

   1. Step 1: Try factors of the constant term until f(a) = 0$\text{A root: } x = 1$$f(1) = (1)^3 - 6(1)^2 + 11(1) - 6$$= 1 - 6 + 11 - 6$$= 0$So $(x - 1)$ is a factor.
   2. Step 2: Write f(x) = (your factor)(x² + px + q), expand the right-hand side and collect like terms$x^3 - 6x^2 + 11x - 6 = (x - 1)(x^2 + px + q)$$(x - 1)(x^2 + px + q) = x^3 + px^2 + qx - x^2 - px - q$$= x^3 + (p - 1)x^2 + (q - p)x - q$
   3. Step 3: Compare the coefficients of x², then the numbers$p - 1 = -6$$p = -5$$-q = -6$$q = 6$Check the x term: $q - p = 6 - (-5)$$q - p = 11$ ✓
   4. Step 4: Factorise the quadratic factor, then set each factor to zero$x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3)$$x = 1$$x = 2$$x = 3$
3. Question 3Exact (surd) roots

   Solve $x^3 + 3x^2 - 2 = 0$, expressing non-integer roots in surd form.

   Show the worked solution

   1. Step 1: Try factors of the constant term until f(a) = 0$\text{A root: } x = -1$$f(-1) = (-1)^3 + 3(-1)^2 - 2$$= -1 + 3 - 2$$= 0$So $(x + 1)$ is a factor.
   2. Step 2: Write f(x) = (your factor)(x² + px + q), expand the right-hand side and collect like terms$x^3 + 3x^2 - 2 = (x + 1)(x^2 + px + q)$$(x + 1)(x^2 + px + q) = x^3 + px^2 + qx + x^2 + px + q$$= x^3 + (p + 1)x^2 + (q + p)x + q$
   3. Step 3: Compare the coefficients of x², then the numbers$p + 1 = 3$$p = 2$$q = -2$Check the x term: $q + p = -2 + 2$$q + p = 0$ ✓
   4. Step 4: The quadratic factor doesn't factorise: use the quadratic formula$x = \dfrac{-2 \pm \sqrt{(2)^2 - 4(1)(-2)}}{2(1)}$$= \dfrac{-2 \pm \sqrt{12}}{2}$
   5. Step 5: Simplify the surd, then write all three solutions$x = -1$$x = -1 - \sqrt{3}$ or $x = -1 + \sqrt{3}$
4. Question 4Sum or difference of two cubes

   Factorise $3x^{3} - 192$ completely.

   Show the worked solution

   1. Step 1: Take out the common factor first$3(x^3 - 64)$
   2. Step 2: Write each term as a cube: a³ and b³$(x)^3 - 4^3$
   3. Step 3: a³ − b³ = (a − b)(a² + ab + b²)$3(x - 4)(x^{2} + 4x + 16)$