Lesson 6: Chapter Summary and Review

New Concept

Let's put our polynomial and factoring skills to the test! This lesson consolidates your ability to manipulate rational expressions—adding, subtracting, multiplying, and dividing them—and use these techniques to solve complex rational equations.

What’s next

Get ready to apply these concepts! You'll tackle a series of interactive examples, practice cards, and challenge problems covering the entire chapter.

Property

A polynomial function has the form

where $a_0, a_1, a_2, \ldots, a_n$ are constants and $a_n \neq 0$. The coefficient $a_n$ of the highest power term is called the lead coefficient.

Property

Examples

• Using $(a+b)^2=a^2+2ab+b^2$, we find $(x+4)^2 = x^2 + 2(x)(4) + 4^2 = x^2 + 8x + 16$.
• Using $(a-b)^2=a^2-2ab+b^2$, we find $(3y-2)^2 = (3y)^2 - 2(3y)(2) + 2^2 = 9y^2 - 12y + 4$.
• Using $(a+b)(a-b)=a^2-b^2$, we find $(5z+1)(5z-1) = (5z)^2 - 1^2 = 25z^2 - 1.

Explanation

These are shortcuts for multiplying binomials that appear often. Memorizing them saves time and helps you recognize patterns when factoring. The last one, the difference of squares, is especially important for simplifying expressions.

Property

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

Examples

• To factor $a^3 + 64$, we identify $x=a$ and $y=4$. Using the sum of cubes formula: $a^3 + 4^3 = (a+4)(a^2 - 4a + 16)$.
• To factor $8m^3 - 1$, we identify $x=2m$ and $y=1$. Using the difference of cubes formula: $(2m)^3 - 1^3 = (2m-1)((2m)^2 + 2m(1) + 1^2) = (2m-1)(4m^2 + 2m + 1)$.
• To factor $27p^3 - 8q^3$, we identify $x=3p$ and $y=2q$. Using the difference of cubes formula: $(3p-2q)(9p^2 + 6pq + 4q^2)$.

Explanation

This is a special factoring rule for expressions with two perfect cube terms. The signs follow a pattern: SOAP (Same, Opposite, Always Positive). This helps you remember the signs in the factored form.

Property

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.

To reduce an algebraic fraction:

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

Examples

• To reduce $\frac{5x+10}{x^2-4}$, first factor: $\frac{5(x+2)}{(x+2)(x-2)}$. Then cancel the common factor $(x+2)$ to get $\frac{5}{x-2}$.
• To reduce $\frac{y^2-y-12}{y^2+5y+6}$, first factor: $\frac{(y-4)(y+3)}{(y+2)(y+3)}$. Then cancel the common factor $(y+3)$ to get $\frac{y-4}{y+2}$.
• To reduce $\frac{6a^3b}{18ab^2}$, identify common factors $6, a, b$: $\frac{6 \cdot a \cdot a \cdot a \cdot b}{3 \cdot 6 \cdot a \cdot b \cdot b}$. Canceling gives $\frac{a^2}{3b}$.

Explanation

Just like with numbers, you can simplify algebraic fractions by canceling out factors that are common to both the top and bottom. Remember to factor everything first! You can only cancel factors (things multiplied), not terms (things added).

Property

To add or subtract algebraic fractions:

1. Find the lowest common denominator (LCD) for the fractions. To do this, factor each denominator completely and include each factor in the LCD as many times as it occurs in any one of the given denominators.
2. Build each fraction to an equivalent one with the LCD.
3. Add or subtract the numerators, and keep the same denominator.
4. Reduce the result if necessary.

Examples

• To calculate $\frac{4}{x-2} + \frac{3}{x+2}$, the LCD is $(x-2)(x+2)$. This gives $\frac{4(x+2)}{(x-2)(x+2)} + \frac{3(x-2)}{(x-2)(x+2)} = \frac{4x+8+3x-6}{(x-2)(x+2)} = \frac{7x+2}{x^2-4}$.
• To calculate $\frac{b}{b^2-9} - \frac{2}{b+3}$, first factor to find the LCD. $\frac{b}{(b-3)(b+3)} - \frac{2}{b+3}$. The LCD is $(b-3)(b+3)$. This becomes $\frac{b}{(b-3)(b+3)} - \frac{2(b-3)}{(b-3)(b+3)} = \frac{b-2b+6}{(b-3)(b+3)} = \frac{-b+6}{b^2-9}$.
• To calculate $\frac{5}{2y} + \frac{3}{y^2}$, the LCD is $2y^2$. This gives $\frac{5(y)}{2y(y)} + \frac{3(2)}{y^2(2)} = \frac{5y}{2y^2} + \frac{6}{2y^2} = \frac{5y+6}{2y^2}$.

Explanation

To add or subtract fractions, they must have the same denominator, just like with numbers. Find the LCD, rewrite each fraction with that denominator, and then combine the numerators. It's all about getting a common ground!

Property

Property of Proportions: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$, as long as $b, d \neq 0$.
Whenever we multiply an equation by an expression containing the variable, we should check that the solution obtained does not cause any of the fractions to be undefined.

Examples

• To solve $\frac{x}{4} = \frac{9}{12}$, cross-multiply to get $12x = 4(9)$, which means $12x = 36$. Dividing by 12 gives $x=3$. This solution is valid.
• To solve $\frac{5}{y-2} = \frac{3}{4}$, cross-multiply: $5(4) = 3(y-2)$. This gives $20 = 3y - 6$, so $26 = 3y$, and $y=\frac{26}{3}$. Checking, the denominator is not zero.
• To solve $\frac{a}{a-5} = \frac{5}{a-5}$, if we cross-multiply, we get $a(a-5) = 5(a-5)$. If we divide by $(a-5)$, we get $a=5$. However, $a=5$ makes the original denominators zero, so it is an extraneous solution. There is no solution.

Explanation

When two fractions are equal, it's called a proportion. You can solve them by 'cross-multiplying.' But be careful! Always check your answer to make sure it doesn't make a denominator zero, which is not allowed.