Chapter 3.3, Problem 1E

If we divide the polynomial P by the factor x − c and we obtain the equation P(x) = (x − c)Q(x) + R(x), then we say that x − c is the divisor, Q(x) is the __________, and R(x) is the __________.

To determine

To fill: The blank in the statement “If we divide the polynomial P by the factor $x-c$ and we obtain the equation $P\left(x\right)=\left(x-c\right)Q\left(x\right)+R\left(x\right),$ then we say that $x-c$ is the divisor, $Q\left(x\right)$ is the ___ , and $R\left(x\right)$ is the ___”.

Answer to Problem 1E

The complete statement is “If we divide the polynomial P by the factor $x-c$ and we obtain the equation $P\left(x\right)=\left(x-c\right)Q\left(x\right)+R\left(x\right),$ then we say that $x-c$ is the divisor, $Q\left(x\right)$ is the $\underset{¯}{\text{quotient}}$, and $R\left(x\right)$ is the $\underset{¯}{\text{remainder}}$”.

Explanation of Solution

Method used:

If $P\left(x\right)$ and $D\left(x\right)$ are polynomials, with $D\left(x\right)\ne 0$, then there exist unique polynomials $Q\left(x\right)$ and $R\left(x\right)$ where $R\left(x\right)$ is either 0 or of degree less than the degree of $D\left(x\right)$, such that $P\left(x\right)=D\left(x\right)\cdot Q\left(x\right)+R\left(x\right)$.

That is, $\text{Dividend}=\text{Divisor}\cdot \text{Quotient}+\text{Remainder}$.

Calculation:

In division, there are four basic terms: dividend, divisor quotient and remainder.

The number which is divided is called the dividend, the number left after the division is called the remainder, the result from the division is called the quotient and the number which divides the dividend is called the divisor.

The given equation is $P(x)=\left(x-c\right)Q\left(x\right)+R\left(x\right)$.

Compare this equation with $\text{Dividend}=\text{Divisor}\cdot \text{Quotient}+\text{Remainder}$.

Here, $Q(x)$ is the quotient and $R\left(x\right)$ is the remainder.

Thus, the complete statement is “If we divide the polynomial P by the factor $x-c$ and we obtain the equation $P\left(x\right)=\left(x-c\right)Q\left(x\right)+R\left(x\right),$ then we say that $x-c$ is the divisor, $Q\left(x\right)$ is the $\underset{¯}{\text{quotient}}$, and $R\left(x\right)$ is the $\underset{¯}{\text{remainder}}$”.