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Chapter 3: Problem 71

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?

Short Answer

Expert verified
The fact that \(\frac{3}{4}\) is a root of the polynomial equation signifies that the balance between the terms in the polynomial will result in it equalling zero when \(x = \frac{3}{4}\). This balance is directly influenced by the leading coefficient and the constant term.

Step by step solution

01

Identify the given polynomial root

The root of the given polynomial is \(\frac{3}{4}\). This means that if we substitute \(\frac{3}{4}\) into our polynomial equation, the result should be zero.
02

Set up a polynomial equation

We can express the polynomial equation as \(ax^n + ... + bx + c = 0\), where 'a' is the leading coefficient, 'c' is the constant term, and n is the degree of the polynomial.
03

Substitute the root into the polynomial

Substitute \(\frac{3}{4}\) into the polynomial equation, we get \(a(\frac{3}{4})^n + ... + b(\frac{3}{4}) + c = 0\)
04

Understand the results

We can't fully solve this, since we don't know the values of a, b, c, or n. But it does tell us that the balance between the terms in the polynomial will result in it equalling zero when \(x = \frac{3}{4}\). This balance is influenced by the leading coefficient (a) and the constant term (c).

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Most popular questions from this chapter

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degreeof \(p\) and \(q\) are as small as possible. More than one correct finction may be possible. Graph your function using a graphing utility to verify that it has the required propertics I has a vertical asymptote given by \(x=1,\) a slant asymptote whose equation is \(y=x, y\) -intercept at \(2,\) and \(x\) -intercepts at \(-1\) and 2

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