Question

Given that one root of \(2 \mathrm{x}^{3}+4 \mathrm{x}^{2}-46 \mathrm{x}-120=0\) is 5 , find 2 more roots.

Short answer

The other two roots of the cubic equation \(2x^3 + 4x^2 - 46x - 120 = 0\) are -3 and -4.

Step-by-step solution

Apply synthetic division

First, we will use synthetic division to divide the given polynomial by \((x-5)\), since 5 is a root. ``` ______________ 5 | 2 4 -46 -120 10 70 580 ______________ 2 14 24 460 ``` The result of synthetic division is \(2x^2+14x+24\).

Solve the quadratic equation

Now, we need to find the roots of the quadratic equation \(2x^2 + 14x + 24 = 0\). We can use the quadratic formula \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\) to find the roots. Here, \(a = 2, b = 14\), and \(c = 24\).

Calculate the discriminant

We calculate the discriminant, which is inside the square root in the quadratic formula: \(\Delta = b^2 - 4ac\). \(\Delta = 14^2 - 4(2)(24) = 196 - 192 = 4\)

Find the other two roots

Since the discriminant is positive, we have two distinct real roots for the quadratic equation. Now, we apply the quadratic formula: \(x = \frac{-14 \pm \sqrt{4}}{4}\) \(x = \frac{-14 \pm 2}{4}\) The two remaining roots are: \(x_1 = \frac{-14 + 2}{4} = \frac{-12}{4} = -3\) \(x_2 = \frac{-14 - 2}{4} = \frac{-16}{4} = -4\) The two other roots of the given cubic equation are -3 and -4.

Key concepts

Understanding Polynomial Roots

The roots of a polynomial are values for which the polynomial equals zero. These are vital for solving various algebraic problems and are particularly important in fields like engineering and physics. In the case of the given cubic polynomial, we've been told that 5 is a root, meaning if we substitute 5 for 'x' in the equation, the polynomial will equal zero. This knowledge allows us to find the remaining roots.

Once we use synthetic division with the known root, we're left with a simpler, quadratic polynomial. Finding the roots of this quadratic equation then gives us the complete set of solutions for the original cubic equation. Remember, the number of roots a polynomial has coincides with its degree, which is the highest power of 'x'. Therefore, a cubic polynomial, such as the one given, will have up to three roots.

Solving the Quadratic Equation

When a polynomial is reduced to a quadratic equation, we can solve it by various methods, including factoring, completing the square, or using the quadratic formula. The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), is particularly useful when factoring is complex or impossible.

This method is reliable because it always works for any quadratic equation where 'a', 'b', and 'c' are real numbers, with 'a' being non-zero. It utilizes the coefficients of the quadratic equation (from \(ax^2 + bx + c = 0\)), substituting them into the formula to find the two roots. In this example, after applying the synthetic division and being left with \(2x^2+14x+24=0\), we plug in the values into the formula to find the remaining roots of the original cubic equation.

The Role of the Discriminant

The discriminant in a quadratic equation, denoted as \(\Delta\), is the part under the square root in the quadratic formula: \(b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of the equation.

The discriminant can tell us whether the roots are real or complex, and whether they are distinct or repeated. Here's a quick reference guide:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), the equation has one repeated real root.
• If \(\Delta < 0\), the equation has two complex roots (which are conjugates of each other).

In our exercise, \(\Delta = 4\), which is greater than zero, indicating that we have two distinct real solutions. Thus, we can proceed with confidence using the quadratic formula to find the two remaining real roots of the equation.