Chapter 3.3, Problem 2E

1. (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.
2. (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) = __________.

To determine

To fill: The blank in the statement “If we divide the polynomial $P\left(x\right)$ by the factor $x-c$ and we obtain a remainder of 0, then we know that c is a _____ of P”.

Answer to Problem 2E

The complete statement is “If we divide the polynomial $P\left(x\right)$ by the factor $x-c$ and we obtain a remainder of 0, then we know that c is a $\underset{¯}{\text{factor}}$ of P”.

Explanation of Solution

Method used:

Division algorithm:

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

Theorem used:

Factor theorem:

c is a zero of P if and only if $x-c$ is a factor of $P\left(x\right)$.

Calculation:

If the remainder is zero then, $P\left(x\right)=D\left(x\right)\cdot Q\left(x\right)+R\left(x\right)$ becomes $P\left(x\right)=D\left(x\right)\cdot Q\left(x\right)$.

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

Here, the polynomial $P\left(x\right)$ is divided by the factor $x-c$.

Then, $P\left(x\right)=\left(x-c\right)\cdot Q\left(x\right)$.

Here, $Q\left(x\right)$ is the quotient and $x-c$ is a factor of $P\left(x\right)$.

Then, by the factor theorem, c is a factor of $P\left(x\right)$.

Thus, the complete statement is “If we divide the polynomial $P\left(x\right)$ by the factor $x-c$ and we obtain a remainder of 0, then we know that c is a $\underset{¯}{\text{factor}}$ of P”.

To determine

To fill: The blank in the statement “If we divide the polynomial $P\left(x\right)$ by the factor $x-c$ and we obtain a remainder of k, then we know that $P\left(c\right)=\_\_\_\_$”.

Answer to Problem 2E

The complete statement is “If we divide the polynomial $P\left(x\right)$ by the factor $x-c$ and we obtain a remainder of k, then we know that $\underset{¯}{P\left(c\right)=k}$”.

Explanation of Solution

If the remainder is k then, $P\left(x\right)=D\left(x\right)\cdot Q\left(x\right)+R\left(x\right)$ becomes $P\left(x\right)=D\left(x\right)\cdot Q\left(x\right)+k$.

Here, the polynomial $P\left(x\right)$ is divided by the factor $x-c$.

Then, $P\left(x\right)=\left(x-c\right)\cdot Q\left(x\right)+k$.

Calculate the value of $P\left(x\right)$,by substituting c for x in the above equation.

$\begin{array}{c}P\left(x\right)=\left(x-c\right)\cdot Q\left(x\right)+k\\ P\left(c\right)=\left(c-c\right)\cdot Q\left(c\right)+k\\ P\left(c\right)=\left(0\right)\cdot Q\left(c\right)+k\\ P\left(c\right)=k\end{array}$

Thus, the complete statement is “If we divide the polynomial $P\left(x\right)$ by the factor $x-c$ and we obtain a remainder of k, then we know that $\underset{¯}{P\left(c\right)=k}$”.