Problem 71 - College Algebra

Question

Suppose \(\frac{3}{4}\) is a root of a polynomial equation. What does this tell us about the leading coefficient and the constant term in the equation?

Step-by-Step Solution

Answer

The leading coefficient of the polynomial is \( a \), and the constant term is \( -a(\frac{3}{4})^{n} \). These two parameters are linked to the root \( \frac{3}{4} \) and the unknown multiplicity \( n \).

Step 1: Apply The Factor Theorem

The Factor Theorem states that if \( r \) is a root of a polynomial \( P(x) \), then \( P(r) = 0 \) and \( (x-r) \) is a factor of that polynomial. In this case, as \( \frac{3}{4} \) is a root, \( (x-\frac{3}{4}) \) is a factor of the given polynomial.

Step 2: Write The General Form of Polynomial

Because \( (x-\frac{3}{4}) \) is a factor of the given polynomial, we can write the general form of this polynomial as \( P(x) = a(x-\frac{3}{4})^{n} \), where \( n \) is the multiplicity of the root \( \frac{3}{4} \), and \( a \) is the leading coefficient. Since we don't have any idea about the degree of the polynomial, this is a general form.

Step 3: Find The Leading Coefficient and Constant Term

The leading coefficient of the polynomial is the coefficient of the highest-degree term, i.e., it's the \( a \) in our polynomial. The constant term of the polynomial is the term that does not contain any variable, i.e. it's the result of the \( P(0) \). Substituting \( x = 0 \) into the equation, the constant term will be \( -a(\frac{3}{4})^{n} \).