Problem 2 - Precalculus Mathematics for Calculus

Question

(a) If we divide the polynomial \(P(x)\) by the factor \(x-c\) and we obtain a remainder of 0 , then we know that \(c\) is a _____ of \(P.\) (b) If we divide the polynomial \(P(x)\) by the factor \(x-c\) and we obtain a remainder of \(k,\) then we know that \(P(c)=\) _____.

Step-by-Step Solution

Answer

(a) Root; (b) k.

Step 1: Understand the Remainder Theorem for Part (a)

The Remainder Theorem states that when a polynomial \(P(x)\) is divided by \(x-c\), the remainder of this division is \(P(c)\). If this remainder is 0, then it means \(P(c) = 0\). This implies \(c\) is a root (or zero) of the polynomial \(P(x)\), meaning \(x-c\) is a factor of \(P(x)\).

Step 2: Answer Part (a)

Based on the understanding from Step 1, if the remainder is 0, then \(c\) is a **root** of \(P(x)\).

Step 3: Apply the Remainder Theorem to Part (b)

The Remainder Theorem indicates that dividing \(P(x)\) by \(x-c\) results in a remainder \(P(c)\). If the remainder of this division is \(k\), then it follows that \(P(c) = k\).

Step 4: Answer Part (b)

Therefore, if the remainder when dividing \(P(x)\) by \(x-c\) is \(k\), then \(P(c) = k\).

Key Concepts

Polynomial Division

Polynomial division is a fundamental concept in algebra, akin to numerical long division but applied to polynomial expressions. It involves dividing a polynomial by another polynomial, usually of a lower degree, to simplify or find particular values. Understanding polynomial division is crucial when working with the Remainder Theorem, roots, and factors of polynomials.
The process typically follows these steps:

• Align the terms of both polynomials by decreasing degree.
• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by the result and subtract it from the dividend.
• Bring down the next term and repeat until the remainder, if any, is less than the degree of the divisor.

This result can be expressed as: \[P(x) = (x-c)Q(x) + R(x)\]where \(P(x)\) is the dividend, \(x-c\) is the divisor, \(Q(x)\) is the quotient, and \(R(x)\) is the remainder. This equation is essential in finding roots and factors of polynomials.

Roots of Polynomials

The roots of a polynomial are the solutions to the equation \(P(x) = 0\). These are the values of \(x\) for which the polynomial evaluates to zero. Finding these roots helps in understanding the behavior of polynomial functions.
The Remainder Theorem aids in identifying these roots by indicating that if a polynomial \(P(x)\) is divided by \(x-c\) and the remainder is zero, then \(c\) is a root of the polynomial. This occurs because \(P(c) = 0\) when \(c\) is a root.
To check for roots:

• Substitute potential root values into the polynomial.
• If the result is zero, the value is a root.

Roots can be real or complex numbers, and polynomials of degree \(n\) have exactly \(n\) roots, counting multiplicities.

Factors of Polynomials

Factors of a polynomial are expressions that divide the polynomial exactly, leaving a remainder of zero. In simpler terms, if \(x-c\) is a factor of \(P(x)\), then \(c\) is a root, and dividing \(P(x)\) by \(x-c\) leaves no remainder.
The relationship between roots and factors is fundamental in algebra:

• If a polynomial has a root \(c\), then \(x-c\) is a factor.
• Conversely, if \(x-c\) is a factor, \(c\) is a root.

Using polynomial division, one can verify if a given expression is a factor by checking if the remainder is zero. The Factor Theorem complements this by stating that for \(P(x)\), \(x-c\) is a factor exactly when \(P(c) = 0\). Thus, identifying factors simplifies problems like polynomial equation solving and graphing.