Question

(a) Find a cubic equation with roots \(6,-1,\) and 3 (b) Find a quartic equation with roots \(1,2,3,\) and 5

Short answer

(a) Cubic: \(x^3 - 8x^2 + 9x + 18 = 0\) (b) Quartic: \(x^4 - 11x^3 + 41x^2 - 61x + 30 = 0\)

Step-by-step solution

Understand the Problem

We need to find a polynomial equation given its roots. In part (a), the roots are 6, -1, and 3. For part (b), the roots are 1, 2, 3, and 5.

Construct Polynomial Equation from Roots (Part a)

For a polynomial equation with roots 6, -1, and 3, we use the fact that if \( r \) is a root, then \( (x - r) \) is a factor. Thus, the factors for part (a) are \((x - 6), (x + 1),\) and \((x - 3)\).

Expand Polynomial (Part a)

Multiply the factors from Step 2: \((x - 6)(x + 1)(x - 3)\).First, multiply \((x - 6)\) and \((x + 1)\):\[(x - 6)(x + 1) = x^2 + x - 6x - 6 = x^2 - 5x - 6\].Next, multiply the result by \((x - 3)\):\[ (x^2 - 5x - 6)(x - 3) = x^3 - 3x^2 - 5x^2 + 15x - 6x + 18 \].Combine like terms:\[ x^3 - 8x^2 + 9x + 18 \].

Construct Polynomial Equation from Roots (Part b)

For a polynomial equation with roots 1, 2, 3, and 5, the factors are \((x - 1), (x - 2), (x - 3),\) and \((x - 5)\).

Expand Polynomial (Part b)

First, multiply \((x - 1)\) and \((x - 2)\):\[(x - 1)(x - 2) = x^2 - 2x - x + 2 = x^2 - 3x + 2\].Then, multiply \((x - 3)\) and \((x - 5)\):\[(x - 3)(x - 5) = x^2 - 5x - 3x + 15 = x^2 - 8x + 15\].Finally, multiply the two quadratics:\[(x^2 - 3x + 2)(x^2 - 8x + 15)\].Expand:\[x^4 - 8x^3 + 15x^2 - 3x^3 + 24x^2 - 45x + 2x^2 - 16x + 30 \].Combine like terms:\[x^4 - 11x^3 + 41x^2 - 61x + 30 \].

Final Polynomial Equations

(a) The cubic equation with roots 6, -1, and 3 is \(x^3 - 8x^2 + 9x + 18 = 0\).(b) The quartic equation with roots 1, 2, 3, and 5 is \(x^4 - 11x^3 + 41x^2 - 61x + 30 = 0\).

Key concepts

Cubic Equations

Cubic equations are polynomial equations of degree three. The general form is \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are coefficients and \(a eq 0\). The task of finding a cubic equation often involves knowing its roots. For example, if the roots are 6, -1, and 3, the factors associated with these roots are \((x - 6)\), \((x + 1)\), and \((x - 3)\). To form the polynomial, you multiply these factors together. This multiplication transforms the factors into the standard form of a cubic equation. This type of equation can have up to three roots, which may be real or complex.

Quartic Equations

Quartic equations are polynomial equations of degree four, with the general form \(ax^4 + bx^3 + cx^2 + dx + e = 0\), where \(a\) is non-zero. Finding a quartic equation when given its roots involves creating linear factors for each root. For instance, if a quartic equation has roots 1, 2, 3, and 5, the corresponding factors are \((x - 1)\), \((x - 2)\), \((x - 3)\), and \((x - 5)\). By multiplying these factors, you arrive at the quartic equation in its expanded form. This multiplication process is crucial because it transitions from a list of roots into a recognized polynomial expression.

Roots of Equations

The roots of an equation are the values of \(x\) that satisfy the equation, making it equal to zero. In polynomial equations, these roots are referred to as the solutions. For cubic equations, there can be up to three roots, and for quartic equations, up to four. The roots can be real numbers or complex numbers. Every polynomial of degree \(n\) is expected to have \(n\) roots, considering multiplicity and complex roots. Roots are crucial for determining the factors of the polynomial because each root corresponds to a factor in the form of \(x - r\), where \(r\) is a root.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. These factors are essential because they break down the polynomial into simpler linear expressions. When given the roots of a polynomial, factoring is straightforward. For instance, with roots \(6, -1,\) and \(3\), the factors are \((x - 6)\), \((x + 1)\), and \((x - 3)\). This approach of converting roots into factors is known as the Factor Theorem. The Factor Theorem states that \((x - r)\) is a factor of a polynomial if and only if \(r\) is a root of the polynomial. Utilizing this theorem can significantly simplify many polynomial operations.