Question

Suppose that the remainder obtained in a polynomial division by \(x-k\) is zero. How is the divisor related to the graph of the dividend?

Short answer

The divisor \(x - k\) corresponds to the roots of the polynomial P(x) and determines where the graph of P(x) intersects the x-axis, which is at \(x = k\).

Step-by-step solution

Identification of Concepts

The exercise is based on polynomial division and the concept of roots or zeros. A polynomial, P(x), divided by \(x - k\) results in a certain remainder. In this case, it is stated that the remainder is zero. This implicates that \(x = k\) is a root or zero of the polynomial P(x). From the Fundamental Theorem of Algebra, we know that a polynomial of degree n has exactly n roots or zeros, counting multiplicity. Therefore, if the remainder obtained in a polynomial division by \(x-k\) is zero, it indicates that the graph of the polynomial cuts the x-axis at \(x = k\).

Implication of a Zero Remainder

If the remainder of a polynomial division is zero, it signifies that the divisor \(x-k\) is a factor of the dividend polynomial, P(x). Thus, \(x=k\) is a root of the polynomial function P(x).

Connection to the Graph of the Polynomial

The root of a polynomial indicates where the graph of the function intersects with the x-axis. Therefore, the divisor \(x - k\) is related to the graph of the dividend polynomial in the sense that it determines where the graph intersects the x-axis, that is, at \(x = k\).

Key concepts

Roots of Polynomials

Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The roots of a polynomial, also known as zeros, are the values of the variable that make the polynomial equal to zero. In simpler terms, when you substitute the root into the polynomial, it yields zero. This is crucial because finding these roots tells us where a polynomial equation intersects the x-axis on a graph.

For example, in the polynomial equation \(f(x) = x^2 - 5x + 6\), the roots are the solutions to \(f(x) = 0\). Graphically, these roots are the x-coordinates where the curve crosses the x-axis. Knowing the roots can help us understand the behavior of the polynomial function across different regions of the graph.

• Roots are solutions to the equation \(P(x) = 0\).
• They indicate the x-intercepts of the polynomial on a graph.
• Multiplicity of a root can affect the shape of the graph at the intercept.

Understanding the roots is essential for solving polynomial equations and analyzing their graphs.

Remainder Theorem

The Remainder Theorem is an important tool in algebra that relates polynomial division to evaluating polynomials. It states that if a polynomial \(P(x)\) is divided by \(x - k\), the remainder of that division is equal to \(P(k)\).

This means you simply need to substitute \(k\) into the polynomial \(P(x)\) to find the remainder, avoiding lengthy division processes. If \(P(k) = 0\), the remainder is zero, indicating that \(x - k\) is a factor of \(P(x)\) and \(k\) is a root of the polynomial. Thus, the Remainder Theorem provides a quick way to confirm if a given value is a root.

• Helps simplify finding remainders in polynomial division.
• Provides a direct link between roots and factors.
• Essential for quickly checking possible roots.

Using this theorem can substantially reduce the effort needed in solving polynomial-related problems.

Factors of Polynomials

Factors of a polynomial are the polynomials that divide the original polynomial without leaving a remainder. If you can express a polynomial \(P(x)\) as a product of other polynomials, then these other expressions are considered factors. For example, if \(P(x) = (x - a)(x - b)\), then \(x - a\) and \(x - b\) are factors of \(P(x)\).

Understanding factors is key to solving polynomial equations and simplifying expressions. Each root \(x = a\) corresponds to a factor \(x - a\). Having the polynomial in its factored form helps in various operations, such as identifying roots, and is central in calculus, particularly when finding limits or solving inequalities.

• Factors give insight into the roots of the polynomial.
• Help in simplifying expressions and solving equations.
• Allow for an easier integration and differentiation in calculus.

Knowing how to factor polynomials is a critical skill for students aiming to master algebraic manipulation and solve polynomial-related problems effectively.