Questions: Solve the higher order inequality and sketch the solution on a number line, then write your answer in interval notation. x^4 + 4x^3 - 21x^2 ≥ 0 *Factor, find the zeros, put them on the number, shade the appropriate regions and then write the solution in interval notation.

Solution Steps

To solve the inequality \(x^{4}+4x^{3}-21x^{2} \geq 0\), we first factor the polynomial to find its roots. These roots will help us determine the critical points on the number line. We then test intervals between these critical points to determine where the inequality holds true. Finally, we express the solution in interval notation.

We start with the polynomial \(x^{4} + 4x^{3} - 21x^{2}\). By factoring, we can express it as: \[ x^{2}(x - 3)(x + 7) \]

Setting the factored polynomial equal to zero gives us the roots: \[ x^{2} = 0 \quad \Rightarrow \quad x = 0 \] \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \] Thus, the roots are \(x = -7\), \(x = 0\), and \(x = 3\).

The critical points divide the number line into intervals: \[ (-\infty, -7), \quad (-7, 0), \quad (0, 3), \quad (3, \infty) \] We will test each interval to see where the inequality \(x^{4} + 4x^{3} - 21x^{2} \geq 0\) holds.

1. For \(x \in (-\infty, -7)\), choose \(x = -8\): \[ (-8)^{4} + 4(-8)^{3} - 21(-8)^{2} = 256 - 1024 - 1344 < 0 \]
2. For \(x \in (-7, 0)\), choose \(x = -1\): \[ (-1)^{4} + 4(-1)^{3} - 21(-1)^{2} = 1 - 4 - 21 < 0 \]
3. For \(x \in (0, 3)\), choose \(x = 1\): \[ (1)^{4} + 4(1)^{3} - 21(1)^{2} = 1 + 4 - 21 < 0 \]
4. For \(x \in (3, \infty)\), choose \(x = 4\): \[ (4)^{4} + 4(4)^{3} - 21(4)^{2} = 256 + 256 - 336 > 0 \]

The inequality holds true in the intervals: \[ (-\infty, -7] \cup [0, 3] \]

Final Answer