Question

Given that \(x=2\) is one root of $$ g(x)=2 x^{4}+4 x^{3}-9 x^{2}-11 x-6=0 $$ use factorisation to determine how many real roots it has.

Short answer

The polynomial \(g(x)\) has three real roots.

Step-by-step solution

- Verify the Given Root

Substitute the given root, \(x = 2\), into the polynomial \(g(x) = 2x^4 + 4x^3 - 9x^2 - 11x - 6\) and verify that it satisfies the equation. That is, calculate \(g(2)\) and check if it equals zero.

- Perform Polynomial Division

Since \(x = 2\) is a root, \(x - 2\) is a factor of \(g(x)\). Divide \(g(x)\) by \(x - 2\) using synthetic or long division to obtain the quotient polynomial.

- Factor the Quotient Polynomial

After finding the quotient polynomial from Step 2, factorize the resulting polynomial further to find its roots. This might involve further polynomial division, factoring by grouping, or using the quadratic formula if necessary.

- Find Real Roots

Determine the real roots of the factorized polynomial. The roots will come from solving each factor set to zero.

- Count the Real Roots

Add the real roots obtained, including the given root \(x=2\), to find the total number of real roots for the polynomial \(g(x)\).

Key concepts

Factorization

To solve polynomial equations, a key method used is factorization. This involves writing the polynomial as a product of simpler polynomials. For example, if we have a polynomial equation like:g(x) = 2x^4 + 4x^3 - 9x^2 - 11x - 6 = 0,and we know that one root is x = 2, then (x - 2) is a factor of g(x). The process of factorization will help break down the polynomial into simpler components, making it easier to solve.

• Identify a root of the polynomial.
• Write the polynomial as a product of `(x - root)` and another polynomial.
• Continue factorizing the resulting polynomial.

By breaking down the polynomial, we simplify the problem and prepare it for further operations like polynomial division or finding real roots.

Polynomial Division

Polynomial division is a crucial step in breaking down polynomials. When we know a root of the polynomial, we can divide the polynomial by its corresponding factor. To divide the polynomial g(x) = 2x^4 + 4x^3 - 9x^2 - 11x - 6 by (x - 2), we can use either long division or synthetic division.

• Place the divisor and dividend in appropriate formats.
• Carry out the division process step-by-step, like elementary number division.
• Obtain the quotient polynomial which can be further investigated.

This step reduces the degree of the polynomial and simplifies the equation, setting up for easier factorization and solving.

Real Roots

Finding real roots of the polynomial is the ultimate goal. A real root is a solution to the polynomial equation that is a real number. After factorizing the polynomial, each factor set to zero gives us potential real roots. For instance, after factorizing g(x) = 2x^4 + 4x^3 - 9x^2 - 11x - 6 using its known root x = 2, solving the resulting factors will provide other real roots.

• Set each factor of the polynomial to zero and solve for x.
• Verify that each solution satisfies the original polynomial equation.
• Count all unique real roots obtained.

Understanding real roots is key in interpreting the solutions to polynomial equations in real-world contexts.

Synthetic Division

Synthetic division offers a quicker alternative to long division when dividing polynomials. It is especially useful for dividing by linear expressions of the form (x - c). For our polynomial g(x) and root x = 2, synthetic division simplifies the process:

• Write down the coefficients of the polynomial.
• Use the root (x = 2) as a divisor in the synthetic division setup.
• Perform the division to obtain the coefficients of the quotient polynomial.

The synthetic division method streamlines polynomial division and helps in quickly finding quotient polynomials.

Quadratic Formula

The quadratic formula is an essential tool for solving second-degree polynomial equations of the form ax^2 + bx + c = 0. After reducing our polynomial using factorization and division, we might end up with a quadratic equation. The quadratic formula, given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]allows us to find the roots of the quadratic polynomial directly.

• Identify coefficients a, b, and c from the quadratic equation.
• Substitute these values into the quadratic formula.
• Simplify to find the real roots (if the discriminant \(b^2 - 4ac\) is non-negative).

Using the quadratic formula offers a straightforward approach to complete the factorization and find all possible roots.