Intermediate Algebra: Functions and Graphs

Subsection Glossary

• polynomial

• degree

• descending powers

• ascending powers

• lead coefficient

• algebraic fraction

• vertical asymptote

• horizontal asymptote

• rational function

• lowest common denominator

• building factor

• like fractions

• common factor

• complex fraction

• proportion

• extraneous solution

Subsection Key Concepts

1. Polynomial Function.

   A polynomial function has the form

   \begin{equation*} f(x) = a_n x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \cdots + a_2x^2 + a_1x + a_0 \end{equation*}

   where \(a_0\text{,}\) \(a_1\text{,}\) \(a_2\text{,}\) \(\ldots\text{,}\) \(a_n\) are constants and \(a_n \ne 0\text{.}\) The coefficient \(a_n\) of the highest power term is called the lead coefficient.

2. Degree of a Product.

   The degree of a product of non-zero polynomials is the sum of the degrees of the factors. That is:

   If \(P(x)\) has degree \(m\) and \(Q(x)\) has degree \(n\text{,}\) then their product \(P(x)Q(x)\) has degree \(m+n\text{.}\)

3. Special Products of Binomials.

   \begin{align*} \amp(a + b)^2 = (a + b) (a + b) = a^2 + 2ab + b^2\\ \amp(a - b)^2 = (a - b) (a - b) = a^2 - 2ab + b^2\\ \amp(a + b) (a - b)= a^2 -b^2 \end{align*}

4. Cube of a Binomial.

   1. \(\displaystyle (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\)
   2. \(\displaystyle (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3\)

5. Factoring the Sum or Difference of Two Cubes.

   1. \(\displaystyle x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)

   2. \(\displaystyle x^3 - y^3 = (x - y)(x^2 + xy + y^2)\)

6. Fundamental Principle of Fractions.

   We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.

   \begin{equation*} \dfrac{a \cdot c}{b \cdot c} = \dfrac{a}{b}~~~~~~\text{if}~~~~~~b,c \ne 0 \end{equation*}

7. When we cancel common factors, we are dividing. Because division is the inverse or opposite operation for multiplication, we can cancel common factors, but we cannot cancel common terms.

8. To reduce an algebraic fraction.

   1. Factor the numerator and the denominator.
   2. Divide the numerator and denominator by any common factors.

9. Exponential Function.

   An exponential function has the form

   \begin{equation*} f(x) = ab^x,~~~~ \text{ where } ~~~b \gt 0 ~~~\text{ and } ~~~b \ne 1 \text{, } ~~~a \ne 0 \end{equation*}

10. Operations on Fractions.

    • If \(b,~d \ne 0\text{,}\) then

      \begin{equation*} \dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd} \end{equation*}

    • If \(b,~c,~d \ne 0\text{,}\) then

      \begin{equation*} \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c} \end{equation*}

    • If \(c \ne 0\text{,}\) then

      \begin{equation*} \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a+b}{c} \end{equation*}
      \begin{equation*} \dfrac{a}{b} - \dfrac{c}{d} = \dfrac{a-b}{c} \end{equation*}

11. To multiply algebraic fractions:.

    1. Factor each numerator and denominator.
    2. If any factor appears in both a numerator and a denominator, divide out that factor.
    3. Multiply the remaining factors of the numerator and the remaining factors of the denominator.
    4. Reduce the product if necessary.

12. To divide algebraic fractions:.

    1. Take the reciprocal of the second fraction and change the operation to multiplication.
    2. Follow the rules for multiplication of fractions.

13. To add or subtract algebraic fractions.

    1. Find the lowest common denominator (LCD) for the fractions.
    2. Build each fraction to an equivalent one with the same denominator.
    3. Add or subtract the resulting like fractions: Add or subtract their numerators, and keep the same denominator.
    4. Reduce the sum or difference if necessary.

14. To Find the LCD.

    1. Factor each denominator completely.
    2. Include each different factor in the LCD as many times as it occurs in any one of the given denominators.

15. To simplify a complex fraction.

    1. Find the LCD of all the fraction contained in the complex fraction.
    2. Multiply the numerator and the denominator of the complex fraction by the LCD.
    3. Reduce the resulting simple fraction, if possible.
16. An algebraic fractrion is "improper" if the degree of the numerator is greater than the degree of the denominator. If it cannot be reduced, we can simplify the expression by treating it as a division of polynomials. The quotient will be the sum of a polynomial and a simpler algebraic fraction.
17. If the equation contains more than one fraction, we can clear all the denominators at once by multiplying both sides by the LCD of the fractions.

18. Property of Proportions.

    \begin{equation*} \blert{\text{If}~~\dfrac{a}{b}=\dfrac{c}{d},~~~\text{then}~~~ad=bc,~~\text{as long as}~ b,d \ne 0} \end{equation*}
19. Whenever we multiply an equation by an expression containing the variable, we should check that the solution obtained does not cause oany of the fractions to be undefined.