Problem 54 - 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 root \(\frac{3}{4}\) means that \(4x - 3\) is a factor of the polynomial. Therefore, the leading coefficient is likely to be 4. The constant term cannot be definitively determined from only this root, but would be -3 if no other factors existed.

Step 1: Identify the Root's Matching Factor

Identify the root and its related factor in the polynomial equation. If \(\frac{3}{4}\) is a root of a polynomial, it implies that the factor \(4x - 3\) is a factor of the polynomial. Sign is reversed in the factor because roots are the values which make the polynomial equal to zero. So, equating \(4x - 3 = 0\), we get \(x = \frac{3}{4}\).

Step 2: Explore Implications for the Leading Coefficient

The leading coefficient of a polynomial is the number that multiplies the variable raised to the highest power. Since \(4x - 3\) is a factor, the leading coefficient in this equation is 4, which multiplies the \(x\) (if the polynomial is expanded).

Step 3: Investigate Influence on Constant Term

The constant term of a polynomial is the term that doesn't contain any variables. In the factors form, this term is usually the product of the individual constant terms in each binomial factor. However, without knowledge of the other factors of this polynomial, we can't definitively determine the constant term. That being said, if \(4x - 3\) is the only factor (it is a linear equation), then the constant term will be -3.