Polynomial Long Division Review

A set of 100 flashcards reviewing polynomial long division, steps, properties, and verification based on the lecture note slide.

What problem is presented in the image?

Find $\frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$ by using long division.

What is the dividend in the expression $\frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$?

$$5x^3 - x^2 + 7x - 2$$

What is the divisor in the expression $\frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$?

$$x^2 + 1$$

What mathematical method is explicitly requested in the problem to perform the division?

Long division

What is the degree of the dividend polynomial $5x^3 - x^2 + 7x - 2$?

$$3$$

What is the degree of the divisor polynomial $x^2 + 1$?

$$2$$

What is the leading coefficient of the dividend $5x^3 - x^2 + 7x - 2$?

$$5$$

What is the leading coefficient of the divisor $x^2 + 1$?

$$1$$

What is the constant term of the dividend $5x^3 - x^2 + 7x - 2$?

$$-2$$

What is the constant term of the divisor $x^2 + 1$?

$$1$$

How can the divisor $x^2 + 1$ be written using a placeholder for the missing linear term?

$$x^2 + 0x + 1$$

How is the expected degree of the quotient calculated prior to performing division?

Subtract the degree of the divisor from the degree of the dividend ($3 - 2 = 1$)

What is the expected degree of the quotient polynomial $Q(x)$?

$$1$$

What is the maximum possible degree of the remainder polynomial $R(x)$ for a quadratic divisor?

$$1$$

In the first step of long division, what two leading terms are divided?

$5x^3$ and $x^2$

What is the result of dividing the leading term $5x^3$ by $x^2$?

$$5x$$

What is the first term of the quotient $Q(x)$?

$$5x$$

What expression is obtained by multiplying the first quotient term $5x$ by the divisor $x^2 + 1$?

$$5x^3 + 5x$$

How is $5x(x^2 + 0x + 1)$ expanded in standard polynomial form?

$$5x^3 + 0x^2 + 5x$$

What subtraction operation is performed in the first step of long division?

$$(5x^3 - x^2 + 7x - 2) - (5x^3 + 5x)$$

What is the result of subtracting $(5x^3 + 5x)$ from $(5x^3 - x^2 + 7x - 2)$?

$$-x^2 + 2x - 2$$

What is the leading term of the first intermediate remainder $-x^2 + 2x - 2$?

$$-x^2$$

What is the degree of the first intermediate remainder $-x^2 + 2x - 2$?

$$2$$

Why does polynomial long division continue after obtaining $-x^2 + 2x - 2$?

Because its degree ($2$) is equal to the degree of the divisor ($2$)

In the second step of division, what two terms are divided?

$-x^2$ and $x^2$

What is the result of dividing $-x^2$ by $x^2$?

$$-1$$

What is the second term of the quotient $Q(x)$?

$$-1$$

What expression is obtained by multiplying the second quotient term $-1$ by the divisor $x^2 + 1$?

$$-x^2 - 1$$

What subtraction operation is performed in the second step of long division?

$$(-x^2 + 2x - 2) - (-x^2 - 1)$$

What is $-x^2 - (-x^2)$ equal to?

$$0$$

What is $2x - 0x$ equal to?

$$2x$$

What is $-2 - (-1)$ equal to?

$$-1$$

What is the result of the second subtraction step $(-x^2 + 2x - 2) - (-x^2 - 1)$?

$$2x - 1$$

What is the degree of the remainder $2x - 1$?

$$1$$

Why does polynomial long division terminate after obtaining $2x - 1$?

Because the degree of $2x - 1$ ($1$) is strictly less than the degree of the divisor $x^2 + 1$ ($2$)

What is the final quotient $Q(x)$ resulting from the long division?

$$5x - 1$$

What is the final remainder $R(x)$ resulting from the long division?

$$2x - 1$$

How is the overall result of the division expressed in fraction form?

$$5x - 1 + \frac{2x - 1}{x^2 + 1}$$

According to the Division Algorithm, how are $P(x)$, $D(x)$, $Q(x)$, and $R(x)$ related?

$$P(x) = D(x)Q(x) + R(x)$$

In the Division Algorithm identity for this problem, what is the product $D(x)Q(x)$?

$$(x^2 + 1)(5x - 1)$$

What is the expanded form of $(x^2 + 1)(5x - 1)$?

$$5x^3 - x^2 + 5x - 1$$

What is the sum of $(5x^3 - x^2 + 5x - 1)$ and the remainder $(2x - 1)$?

$$5x^3 - x^2 + 7x - 2$$

Does the calculated sum $D(x)Q(x) + R(x)$ equal the original dividend $P(x)$?

Yes, $5x^3 - x^2 + 7x - 2 = 5x^3 - x^2 + 7x - 2$

Why can synthetic division NOT be directly applied to this division problem?

Synthetic division requires a linear divisor of the form $x - c$, whereas $x^2 + 1$ is quadratic.

What is the domain of the rational function $f(x) = \frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$?

All real numbers, or $(-\infty, \infty)$

Why does the rational function $f(x) = \frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$ have no real vertical asymptotes?

The denominator $x^2 + 1$ has no real zeros because $x^2 + 1 > 0$ for all real $x$.

What type of non-vertical asymptote does the graph of $f(x) = \frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$ possess?

A slant (oblique) asymptote

What is the equation of the slant asymptote for $f(x) = \frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$?

$$y = 5x - 1$$

Why is $y = 5x - 1$ the slant asymptote of the function $f(x)$?

Because as $x \rightarrow \pm\infty$, the remainder term $\frac{2x - 1}{x^2 + 1}$ approaches $0$.

What is the $y$-intercept of the rational function $f(x) = \frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$?

$$(0, -2)$$

What is the value of $f(0)$ calculated directly from the original rational expression?

$$\frac{5(0)^3 - (0)^2 + 7(0) - 2}{0^2 + 1} = -2$$

What is the value of $f(0)$ calculated using the quotient plus remainder expression $5x - 1 + \frac{2x - 1}{x^2 + 1}$?

$$5(0) - 1 + \frac{2(0) - 1}{0^2 + 1} = -1 + (-1) = -2$$

What is the value of the dividend $P(x) = 5x^3 - x^2 + 7x - 2$ evaluated at $x = 1$?

$$9$$

What is the value of the divisor $D(x) = x^2 + 1$ evaluated at $x = 1$?

$$2$$

What is the value of the quotient $Q(x) = 5x - 1$ evaluated at $x = 1$?

$$4$$

What is the value of the remainder $R(x) = 2x - 1$ evaluated at $x = 1$?

$$1$$

What is the value of $Q(1) + \frac{R(1)}{D(1)}$?

$$4 + \frac{1}{2} = \frac{9}{2}$$

What is the value of the original fraction at $x = 1$?

$$\frac{9}{2}$$

What is the value of the dividend $P(x)$ evaluated at $x = 2$?

$$48$$

What is the value of the divisor $D(x)$ evaluated at $x = 2$?

$$5$$

What is the value of the quotient $Q(x)$ evaluated at $x = 2$?

$$9$$

What is the value of the remainder $R(x)$ evaluated at $x = 2$?

$$3$$

What is the value of $Q(2) + \frac{R(2)}{D(2)}$?

$$9 + \frac{3}{5} = \frac{48}{5}$$

What is the value of the original fraction at $x = 2$?

$$\frac{48}{5}$$

What is the value of the dividend $P(x)$ evaluated at $x = -1$?

$$-15$$

What is the value of the divisor $D(x)$ evaluated at $x = -1$?

$$2$$

What is the value of the quotient $Q(x)$ evaluated at $x = -1$?

$$-6$$

What is the value of the remainder $R(x)$ evaluated at $x = -1$?

$$-3$$

What is the value of $Q(-1) + \frac{R(-1)}{D(-1)}$?

$$-6 + \frac{-3}{2} = -\frac{15}{2}$$

What is the value of the original fraction at $x = -1$?

$$-\frac{15}{2}$$

What are the complex roots of the divisor $x^2 + 1$?

$x = i$ and $x = -i$

What is the value of the remainder $R(x) = 2x - 1$ evaluated at $x = i$?

$$2i - 1$$

What is the value of the dividend $P(x) = 5x^3 - x^2 + 7x - 2$ evaluated at $x = i$?

$$2i - 1$$

What polynomial identity explains why $P(i) = R(i)$?

Since $P(x) = D(x)Q(x) + R(x)$ and $D(i) = 0$, $P(i) = 0 \cdot Q(i) + R(i) = R(i)$.

What is the value of the dividend $P(x)$ evaluated at $x = -i$?

$$-2i - 1$$

What is the value of the remainder $R(x)$ evaluated at $x = -i$?

$$-2i - 1$$

In the division identity $\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$, what is $P(x)$?

$$5x^3 - x^2 + 7x - 2$$

In the division identity $\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$, what is $D(x)$?

$$x^2 + 1$$

In the division identity $\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$, what is $Q(x)$?

$$5x - 1$$

In the division identity $\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$, what is $R(x)$?

$$2x - 1$$

What is the coefficient of $x$ in the quotient $Q(x) = 5x - 1$?

$$5$$

What is the constant term of the quotient $Q(x) = 5x - 1$?

$$-1$$

What is the coefficient of $x$ in the remainder $R(x) = 2x - 1$?

$$2$$

What is the constant term of the remainder $R(x) = 2x - 1$?

$$-1$$

What is the sum of the quotient $Q(x) = 5x - 1$ and the remainder $R(x) = 2x - 1$?

$$7x - 2$$

What is the difference between the quotient $Q(x) = 5x - 1$ and the remainder $R(x) = 2x - 1$?

$$3x$$

What is the product of the quotient $Q(x) = 5x - 1$ and the remainder $R(x) = 2x - 1$?

$$10x^2 - 7x + 1$$

What is the term of degree $3$ in the dividend $5x^3 - x^2 + 7x - 2$?

$$5x^3$$

What is the term of degree $2$ in the dividend $5x^3 - x^2 + 7x - 2$?

$$-x^2$$

What is the term of degree $1$ in the dividend $5x^3 - x^2 + 7x - 2$?

$$7x$$

What is the term of degree $0$ in the dividend $5x^3 - x^2 + 7x - 2$?

$$-2$$

What is the term of degree $2$ in the divisor $x^2 + 1$?

$$x^2$$

What is the term of degree $1$ in the divisor $x^2 + 1$?

$$0x$$

What is the term of degree $0$ in the divisor $x^2 + 1$?

$$1$$

What is the limit of $\frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$ as $x \rightarrow \infty$?

$$\infty$$

What is the limit of $\frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$ as $x \rightarrow -\infty$?

$$-\infty$$

What is the limit of $\frac{f(x)}{5x}$ as $x \rightarrow \infty$ for $f(x) = \frac{5x^3 - x^2 + 7x - 2}{x^2 + 1}$?

$$1$$

At what real value of $x$ does the remainder $R(x) = 2x - 1$ equal zero?

$$x = \frac{1}{2}$$

What does a non-zero remainder $R(x)$ indicate about the relationship between the divisor $D(x)$ and the dividend $P(x)$?

It indicates that the divisor $D(x)$ is not a factor of the dividend $P(x)$.

Is $x^2 + 1$ a factor of $5x^3 - x^2 + 7x - 2$?

No, because the remainder $2x - 1$ is non-zero.