Chapter 5, Problem 4CV

Solution

To match : The vocabulary term with the description that best fits it.

1-D, 2-B, 3-C, 4-A.

Given information :

1. Conjugate Root Theorem 2. Fundamental Theorem of Algebra 3. Rational Root Theorem 4. Remainder Theorem

A. Determines $P\left(a\right)$ by dividing the polynomial by $x-a$ B. The degree equals the number of roots. C. Minimizes guessing fraction and integer solutions. D. Complex numbers as roots come in pairs.

Explanation :

According to the Conjugate Root Theorem, if a root of a polynomial is a complex number, say $a+bi$ , then its complex conjugate, $a-bi$ , will also be a root. This means that the conjugate roots come in pairs.

According to theFundamental Theorem of Algebra, the degree of a polynomial equation tells the number of roots an equation will have. So, if $P\left(x\right)$ is a polynomial of degree $2$ , then $P\left(x\right)=0$ will have exactly $2$ roots and it includes multiple and complex roots.

According to theRational Root Theorem, if the coefficient of a polynomial is integers then it becomes easy to find all the possible rational roots. The roots can be found by dividing each factor of the constant term by each factor of the leading coefficient. It is used to find the rational solution of a polynomial equation.

According to the Remainder theorem, when a polynomial $P\left(a\right)$ is divided by the polynomial $\left(x-a\right)$ then the remainder is $f\left(a\right)$ .

So,

  

Therefore,

1-D, 2-B, 3-C, 4-A.