Precalculus Test 3 Review – Step-by-Step Study Guidance

Q1. Use polynomial long division to rewrite the following fraction in the form $q(x) + \frac{r(x)}{d(x)}$, where $d(x)$ is the denominator, $q(x)$ is the quotient, and $r(x)$ is the remainder:

$\frac{x^3 - 5x^2 + 5x - 3}{x^2 - x}$

Background

Topic: Polynomial Long Division

This question tests your ability to divide polynomials using long division, expressing the result as a sum of a polynomial and a proper fraction.

Key Terms and Formulas

• Dividend: The numerator polynomial ($x^3 - 5x^2 + 5x - 3$)

• Divisor: The denominator polynomial ($x^2 - x$)

• Quotient: The result of the division

• Remainder: What is left after division

• Long Division Algorithm for Polynomials

Step-by-Step Guidance

1. Arrange both the dividend and divisor in descending powers of $x$.

2. Divide the leading term of the dividend ($x^3$) by the leading term of the divisor ($x^2$) to find the first term of the quotient.

3. Multiply the entire divisor by this term and subtract the result from the dividend to get a new polynomial.

4. Repeat the process with the new polynomial: divide its leading term by the leading term of the divisor, multiply, and subtract.

5. Continue until the degree of the remainder is less than the degree of the divisor.

Try solving on your own before revealing the answer!

Final Answer: $3x - 5 + \frac{2x - 3}{x^2 - x}$

After performing the division, the quotient is $3x - 5$ and the remainder is $2x - 3$.

Q2. Use polynomial long division to rewrite the following fraction in the form $q(x) + \frac{r(x)}{d(x)}$:

$\frac{2x^4 + 3x^3 - 3x^2 - 10x + 8}{x^2 - x}$

Background

Topic: Polynomial Long Division

This question is similar to Q1 and tests your ability to perform long division with higher-degree polynomials.

Key Terms and Formulas

• Dividend: $2x^4 + 3x^3 - 3x^2 - 10x + 8$

• Divisor: $x^2 - x$

• Quotient and Remainder

Step-by-Step Guidance

1. Write both polynomials in standard form (descending powers of $x$).

2. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.

3. Multiply the divisor by this term and subtract from the dividend.

4. Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the divisor.

Try solving on your own before revealing the answer!

Final Answer: $2x^2 + 3x - 3 + \frac{-7x + 8}{x^2 - x}$

The quotient is $2x^2 + 3x - 3$ and the remainder is $-7x + 8$.

Q3. Use synthetic division to find the remaining zeros of $f(x) = x^3 + 4x^2 + x - 6$ given that $-2$ is a zero.

Background

Topic: Synthetic Division and Polynomial Zeros

This question tests your ability to use synthetic division to factor a polynomial and find all its zeros, given one zero.

Key Terms and Formulas

• Synthetic Division: A shortcut for dividing a polynomial by a linear factor $(x - c)$.

• Zero of a Polynomial: A value $x = a$ such that $f(a) = 0$.

Step-by-Step Guidance

1. Set up synthetic division using $c = -2$ and the coefficients of $f(x)$: $[1, 4, 1, -6]$.

2. Carry out the synthetic division process to obtain the coefficients of the depressed quadratic.

3. Write the resulting quadratic equation.

4. Factor or use the quadratic formula to find the remaining zeros.

Try solving on your own before revealing the answer!

Final Answer: The remaining zeros are $-3$ and $1$.

After synthetic division, the quadratic factors as $(x + 3)(x - 1)$.

Q4. Use synthetic division to find the remaining zeros of $f(x) = x^3 + 2x^2 - 11x - 12$ given that $3$ is a zero.

Background

Topic: Synthetic Division and Polynomial Zeros

This question is similar to Q3 and tests your ability to use synthetic division to find all zeros of a cubic polynomial.

Key Terms and Formulas

• Synthetic Division

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step-by-Step Guidance

1. Set up synthetic division with $c = 3$ and the coefficients $[1, 2, -11, -12]$.

2. Perform synthetic division to get the depressed quadratic.

3. Write the quadratic equation and solve for its zeros.

Try solving on your own before revealing the answer!

Final Answer: The remaining zeros are $-1$ and $-4$.

After synthetic division, the quadratic factors as $(x + 1)(x + 4)$.

Q5. Find the equations of the (a) vertical, (b) horizontal, and (c) slant asymptotes of $f(x) = \frac{x^2 - 5x + 6}{x - 3}$.

Background

Topic: Rational Functions and Asymptotes

This question tests your understanding of how to find vertical, horizontal, and slant (oblique) asymptotes for rational functions.

Key Terms and Formulas

• Vertical Asymptote: Set denominator $= 0$ and solve for $x$.

• Horizontal Asymptote: Compare degrees of numerator and denominator.

• Slant Asymptote: If degree of numerator is one more than denominator, use polynomial division.

Step-by-Step Guidance

1. Set the denominator $x - 3 = 0$ to find the vertical asymptote.

2. Compare the degrees of numerator and denominator to determine if a horizontal or slant asymptote exists.

3. If the degree of the numerator is greater by one, perform polynomial division to find the slant asymptote.

Try solving on your own before revealing the answer!

Final Answer:

• (a) Vertical asymptote: $x = 3$

• (b) Horizontal asymptote: $y = 1$

• (c) Slant asymptote: none