Question

Statement 1 Roots of the equation \(4 x^{3}+7 x^{2}+7 x+4=0\) are \(-1\) and \(\frac{3 \pm \sqrt{55}}{8}\). and Statement 2 \(\mathrm{x}=-1\) is a root of the cubic equation \(\mathrm{ax}^{3}+\mathrm{bx}^{2}+\mathrm{bx}+\mathrm{a}=0\).

Short answer

Statement 1: The roots of the equation \(4x^3 + 7x^2 + 7x + 4 = 0\) are -1 and \(\frac{3 \pm \sqrt{55}}{8}\). Statement 2: x = -1 is a root of the equation \(ax^3 + bx^2 + bx + a = 0\). Answer: Statement 1 is false, and Statement 2 is true.

Step-by-step solution

Find the roots of the given equation in Statement 1

We need to find the roots of the equation \(4x^3 + 7x^2 + 7x + 4 = 0\). But, as per the statement, we already have the roots as \(-1\) and \(\frac{3 \pm \sqrt{55}}{8}\). We will verify whether these are the correct roots or not.

Use the Rational Root Theorem to verify the roots

In order to verify whether these roots are really correct, we can use the Rational Root Theorem. Since the coefficients of the polynomial in Statement 1 are integers, if a rational number r/s is a root, then r is a divisor of the constant term 4 and s is a divisor of the leading coefficient 4. Therefore, the only possible rational roots for this polynomial are \(\pm 1\), \(\pm 2\), and \(\pm 4\). We will test these values. Testing x = -1: \(4(-1)^3 + 7(-1)^2 + 7(-1) + 4 = -4 + 7 - 7 + 4 = 0\) Since the polynomial equals 0 when x = -1, -1 is indeed a root. So, the first root is correct. Now, let's find the other roots.

Using Synthetic Division to find the remaining roots

As we know x = -1 is a root, let's use synthetic division to find the remaining quadratic equation. We will have: ``` 4 7 7 4 -1 |_________ -4 1 -6 __________________ 4 3 8 -2 ``` So, the remaining quadratic equation is \(4x^2 + 3x + 8 = 0\).

Solving the quadratic equation for the other roots

To find the other roots, we will use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}\). In this case, \(a = 4\), \(b = 3\), and \(c = 8\). \(x = \frac{-3 \pm \sqrt{3^2 -4(4)(8)}}{2(4)} = \frac{-3 \pm \sqrt{9 - 128}}{8} = \frac{-3 \pm \sqrt{-119}}{8}\) Hence, the other roots are \(\frac{-3 \pm \sqrt{-119}}{8}\). So, statement 1 is false as the given roots are incorrect.

Check if x = -1 is the root of the equation in Statement 2

Now, we need to check if x = -1 is a root of the equation \(ax^3 + bx^2 + bx + a = 0\). Plugging in x = -1, we get: \(-a + b - b + a = 0\) Since the given equation simplifies to 0, x = -1 is indeed a root of the given equation. Hence, statement 2 is true. In conclusion, Statement 1 is false, and Statement 2 is true.

Key concepts

Rational Root Theorem

The Rational Root Theorem is a handy mathematical tool used to identify possible rational roots of a polynomial equation. This theorem applies specifically to polynomials with integer coefficients. It helps us narrow down potential root options significantly, which is immensely useful when solving polynomial equations.

Here's how the theorem works:

• The possible rational roots of a polynomial are given by the factors of the constant term divided by the factors of the leading coefficient.
• For example, in the polynomial equation \(4x^3 + 7x^2 + 7x + 4 = 0\), the constant term is 4, and the leading coefficient is also 4.
• Thus, the potential rational roots are \( \pm 1\), \( \pm 2\), and \( \pm 4\).

By testing each of these rational candidates in the polynomial equation, we can confirm if they are indeed roots. This process provides a starting point for verifying roots that might not be obvious at first.

Synthetic Division

Synthetic division is a streamlined method of dividing polynomials, which is particularly useful when checking potential roots of polynomial equations. Compared to the long division of polynomials, synthetic division is quicker because it only requires numerical calculations without the use of variables.

To perform synthetic division:

• Firstly, identify a root, say \(x = -1\), and set up the coefficients of the polynomial.
• Write these coefficients in a row: for \(4x^3 + 7x^2 + 7x + 4\), you have \(4, 7, 7, 4\).
• Bring down the first coefficient directly. Multiply the root by this number and add it to the next coefficient.
• Continue this process across the row of coefficients.
• In this case, the outcome reveals a remainder of zero, confirming \(-1\) as a true root. Thus, forming a new quadratic equation \(4x^2 + 3x + 8 = 0\).

Synthetic division not only helps verify roots but also reduces higher-degree polynomials to manageable lower degrees, essential for simplifying further calculations.

Quadratic Formula

The quadratic formula is an essential formula used to find the roots of a quadratic equation. It applies to any quadratic equation of the form \(ax^2 + bx + c = 0\). This formula is particularly useful when a quadratic does not factor easily or when completing the square becomes cumbersome.

The quadratic formula is expressed as follows:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To find the roots using this formula:

• Identify coefficients \(a\), \(b\), and \(c\) in your quadratic equation. For \(4x^2 + 3x + 8 = 0\), these are \(a = 4\), \(b = 3\), and \(c = 8\).
• Substitute these values into the quadratic formula.
• Compute the discriminant \(b^2 - 4ac\). In this case, the discriminant is negative \( (9 - 128) = -119 \), indicating complex roots.
• Follow through the formula to arrive at \(x = \frac{-3 \pm \sqrt{-119}}{8}\).

This process clarifies why an equation might have non-real roots if the discriminant is less than zero. Hence, the quadratic formula is vital in determining not just real roots, but also complex numbers which could be the equation's solutions.