Problem 44 - Algebra 2

Question

Three roots of a polynomial equation with rational coefficients are \(5+\sqrt{3}\) , \(-17,\) and \(2-\sqrt{4}\) . Which number also is a root of the equation? \(\begin{array}{llll}{\text { A. } 17} & {\text { B. } 2+\sqrt{4}} & {\text { C. } 4-\sqrt{2}} & {\text { D. } 5-\sqrt{3}}\end{array}\)

Step-by-Step Solution

Answer

The number 5−√3 is also a root of the equation.

Step 1: Understand the Concept of Conjugate Roots

If a polynomial has rational coefficients and a root of the form a+b√c, where a and b are rational numbers and c is a positive integer that is not a perfect square, then its conjugate a−b√c must also be a root of the polynomial.

Step 2: Identify the Conjugate Root

The given root 5+√3 suggests that its conjugate 5−√3 must also be a root of the polynomial since the coefficients of the polynomial are rational.

Step 3: Match the Conjugate Root to the Options

Compare the conjugate 5−√3 to the given options to find the match. Option D, 5−√3, is the conjugate root.

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions that involve variables raised to whole number powers and coefficients, which may be numbers or functions of some variables. They are fundamental in algebra and appear in a variety of contexts from basic algebra problems to more complex aspects of mathematics and applied science.

A polynomial equation can typically be written in the form of \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0\), where \(a_n, a_{n-1}, ..., a_0\) are coefficients, \(x\) is the variable, and \(n\) is a non-negative integer representing the degree of the polynomial. The highest power of the variable determines the degree of the polynomial equation.

For example, a quadratic polynomial equation has a degree of 2 and usually appears as \(ax^2 + bx + c = 0\), where a, b, and c are coefficients. The degree of the polynomial equation is vital because it determines the maximum number of roots (solutions) that the equation can have. According to the Fundamental Theorem of Algebra, a polynomial equation of degree \(n\) has precisely \(n\) roots, though some may be repeated or non-real if we consider complex numbers.

Rational Coefficients

Coefficients in a polynomial equation are considered rational if they can be expressed as fractions of integers. That means for every coefficient \(a_i\) in a polynomial \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0\), we can represent it in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \eq 0\).

Having rational coefficients is especially interesting when considering the nature of the roots of the polynomial equation. This is due to the Rational Root Theorem, which states that any rational solution \(\frac{p}{q}\), where p and q are coprime, of the polynomial equation with rational coefficients must be a divisor of the constant term \(a_0\), when divided by a divisor of the leading coefficient \(a_n\).

This property is essential for narrowing down the possible rational roots of the equation. Nevertheless, not all roots of an equation with rational coefficients need to be rational numbers; they can be irrational or complex numbers as well.

Irrational Roots

Irrational roots of polynomial equations are numbers that cannot be expressed as simple fractions, unlike rational numbers. They usually contain square roots, cube roots, or other higher roots of non-perfect powers. An important feature of polynomials with rational coefficients is that these irrational roots come in pairs known as conjugate roots.

Conjugate roots are based on the fact that if a polynomial has rational coefficients and one of its roots is of the form \(a + b\sqrt{c}\), where \(b\) is nonzero and \(c\) is not a perfect square, then it must have a conjugate root of the form \(a - b\sqrt{c}\). This pair of roots ensures that when the polynomial is multiplied out or expanded, the irrational parts cancel each other out, leaving an expression with rational coefficients.

Recognizing pairs of irrational conjugate roots is very useful when solving polynomial equations, as identifying one irrational root immediately tells us that there is another root, which is its conjugate. This principle also helps us understand the symmetry typically involved in equations with rational coefficients and their solutions.