The Language of Algebra: Chapter Review

Vocabulary and notation

Variable

A letter standing for a number: one particular unknown to be found, as in $x + 3 = 7$, or every number at once, as in $a + b = b + a$.

Expression and equation

An expression names a value and has no equals sign, so you evaluate or simplify it; an equation joins two with an equals sign, so you solve it.

Identity

An equation true for every value of its variables, such as $a + 0 = a$. A conditional equation like $x + 3 = 7$ holds only at particular values.

Equivalent expressions

Two expressions giving the same value for every input, so they are exactly the two sides of an identity.

Counterexample

A single value at which the two sides of a claimed identity disagree. One disproves the claim; no finite list of agreements can prove one.

Term, coefficient, and variable part

A term is a signed piece of an expression added to the others; its coefficient is the numerical factor, its variable part the rest. In $-5x$ these are $-5$ and $x$; a lone $x$ has coefficient $1$.

Like terms

Terms with the same variable part: the same letters at the same powers. So $4xy$ and $7yx$ are like terms, $3x^2$ and $5x$ are not.

Index and radicand of $\sqrt[n]{a}$

In $\sqrt[n]{a}$ the index is $n$ and the radicand is $a$. A square root is the case $n = 2$, with the index left unwritten.

Like radicals

Radicals sharing both an index and a radicand, such as $2\sqrt{3}$ and $5\sqrt{3}$.

Greatest common factor (GCF)

The largest factor every term shares: the greatest common divisor of the coefficients times each common variable at its lowest power. For $4x^2 + 6x$ it is $2x$.

Algebraic fraction (rational expression)

A ratio of two expressions, such as $\dfrac{x + 1}{x - 2}$. The bar still means divide top by bottom.

Excluded value

A value of the variable making a denominator zero, where the fraction is undefined: $\dfrac{x}{2x - 6}$ excludes $x = 3$.

Least common denominator (LCD)

The smallest expression every denominator divides into. Adding or subtracting needs a common denominator, and the LCD is the one that keeps the work smallest. For $\dfrac{3}{x}$ and $\dfrac{1}{2}$ it is $2x$.

Formulas and theorems

• The laws of arithmetic, as identities

  $$\begin{gathered} a + b = b + a \\ ab = ba \\ (a + b) + c = a + (b + c) \\ (ab)c = a(bc) \\ a + 0 = a \\ a \cdot 1 = a \\ a + (-a) = 0 \\ a \cdot \tfrac{1}{a} = 1 \end{gathered}$$

  Use when Every real number, with the single restriction $a \neq 0$ in the multiplicative inverse. Subtraction and division are neither commutative nor associative: $5 - 3 \neq 3 - 5$ and $8 \div 2 \neq 2 \div 8$.

• The distributive law

  $$a(b + c) = ab + ac$$

  Text description

  A rectangle of height a and width b plus c, cut once into a piece of area a times b and a piece of area a times c.

  Use when Any real numbers, and any number of terms inside: $a(b + c + d) = ab + ac + ad$. The factor reaches every term, sign included.

  e.g. $-3(x - 4) = -3x + 12$.

• Combining like terms

  $$mv + nv = (m + n)v$$

  Use when Only when the variable part $v$ is identical in both terms; this is the distributive law read in reverse, and it combines like radicals too, with the shared radical playing the part of $v$.

  e.g. $4x^2 - 2x^2 = 2x^2$, while $2x^2 + 3x$ is already finished.

• Subtracting a whole expression

  $$-(a + b) = -a - b$$

  Use when Any expression. A leading minus is the factor $-1$ waiting to be distributed, so it flips every term's sign, not only the first.

  e.g. $(7x + 2) - (3x - 5) = 7x + 2 - 3x + 5 = 4x + 7$.

• Product of two expressions

  $$(a + b)(c + d) = ac + ad + bc + bd$$

  Use when Written for two two-term factors, whose four products (First, Outer, Inner, Last) give the mnemonic FOIL. Longer factors need the general rule instead: every term of the first times every term of the second, then combine like terms.

  e.g. $(2x - 1)(x + 4) = 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4$.

• Two special products

  $$\begin{gathered} (a + b)(a - b) = a^2 - b^2 \\ (a + b)^2 = a^2 + 2ab + b^2 \end{gathered}$$

  Text description

  Square of side (a+b) cut into a², two equal ab rectangles, b².

  Use when Neither is a separate rule: both are the same every-term-times-every-term expansion. In the first the two middle terms cancel; in the second they double.

  e.g. $(x + 5)(x - 5) = x^2 - 25$, while $(x + 3)^2 = x^2 + 6x + 9$.

• Factoring out a common factor

  $$ab + ac = a(b + c)$$

  Use when $a$ must be a factor of every term. Fully factored means $a$ is the GCF.

  e.g. $12x^2 - 8x = 4x(3x - 2)$.

• Unit-fraction exponent: the $n$th root

  $$a^{1/n} = \sqrt[n]{a}$$

  Use when $n$ a positive integer. Even $n$ needs $a \ge 0$ and gives the non-negative (principal) root; odd $n$ allows a negative $a$, with one negative root.

  e.g. $64^{1/3} = \sqrt[3]{64} = 4$, and $\sqrt[3]{-8} = -2$.

• General fractional exponent

  $$a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$$

  Use when Same domain rules as $a^{1/n}$: $n$ a positive integer, and even $n$ needs $a \ge 0$. Rooting first keeps the numbers small.

  e.g. $32^{3/5} = \left(\sqrt[5]{32}\right)^3 = 2^3 = 8$.

• Negative fractional exponent

  $$a^{-m/n} = \frac{1}{a^{m/n}}$$

  Use when $a \neq 0$, and $a > 0$ when $n$ is even. The minus sign moves the power into the denominator.

  e.g. $16^{-3/4} = \dfrac{1}{\left(\sqrt[4]{16}\right)^3} = \dfrac{1}{8}$.

• Product and quotient rules for radicals

  $$\sqrt[n]{ab} = \sqrt[n]{a}\,\sqrt[n]{b}, \qquad \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$$

  Use when One index throughout, $b \neq 0$ in the quotient, and radicands zero or positive when $n$ is even. Roots split across products and quotients, never across a sum.

  e.g. $\sqrt{72} = \sqrt{36}\,\sqrt{2} = 6\sqrt{2}$, and $\dfrac{\sqrt{50}}{\sqrt{2}} = \sqrt{25} = 5$.

• Cancelling a common factor

  $$\frac{ac}{bc} = \frac{a}{b}$$

  Use when $b \neq 0$ and $c \neq 0$, and $c$ must be a factor of the whole numerator and whole denominator, never one term of a sum.

  e.g. $\dfrac{6x^2}{9x} = \dfrac{3x \cdot 2x}{3x \cdot 3} = \dfrac{2x}{3}$ for $x \neq 0$.

• Multiplying and dividing algebraic fractions

  $$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}, \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$

  Use when $b \neq 0$ and $d \neq 0$; division also needs $c \neq 0$. No common denominator is involved.

  e.g. $\dfrac{x}{4} \div \dfrac{x^2}{8} = \dfrac{x}{4} \cdot \dfrac{8}{x^2} = \dfrac{2}{x}$ for $x \neq 0$.

• Adding and subtracting over a common denominator

  $$\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d}, \qquad \frac{a}{d} - \frac{b}{d} = \frac{a - b}{d}$$

  Use when $d \neq 0$, and the denominators must be made equal first, over the LCD.

  e.g. $\dfrac{5x}{x + 1} + \dfrac{2x}{x + 1} = \dfrac{7x}{x + 1}$.

Problem types, step by step

Exam traps

• Trap Cancelling a term rather than a factor: $\dfrac{x + 3}{3} = x + 1$.

  Fix Only a factor of the whole top and bottom cancels, and the $3$ on top is added, not multiplied. The honest split is $\dfrac{x}{3} + 1$: at $x = 3$ the fraction is $2$, not $4$.

• Trap Splitting a root across a sum, or merging unlike radicals: $\sqrt{9 + 16} = 3 + 4$, or $\sqrt{2} + \sqrt{3} = \sqrt{5}$.

  Fix Roots split across products and quotients only: $\sqrt{9 + 16} = \sqrt{25} = 5$, not $7$, and $\sqrt{2} + \sqrt{3}$ does not combine.

• Trap Squaring a sum term by term: $(x + 3)^2 = x^2 + 9$.

  Fix $(x + 3)^2$ is the product $(x + 3)(x + 3)$, whose two cross products supply the middle $6x$.

• Trap Dropping the restriction once a factor cancels, because the simplified form looks harmless.

  Fix Excluded values come from the original denominator and survive cancelling, so $\dfrac{3x + 6}{x + 2} = 3$ still requires $x \neq -2$.

• Trap Treating agreement at a few substituted values as proof that an equation is an identity.

  Fix A finite list leaves infinitely many values untested, so agreement only raises confidence; certainty comes only from a derivation by the laws.

• Trap Swapping the roles of $m$ and $n$ in $a^{m/n}$.

  Fix The denominator is always the root index and the numerator the power, so $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 4$, not $\left(\sqrt{8}\right)^3$.

• Trap Reading a negative exponent as a negative answer: $8^{-2/3} = -4$.

  Fix The minus sign builds a reciprocal rather than changing a sign, so $8^{-2/3} = \dfrac{1}{4}$.

• Trap Stopping at a partial common factor: $8x^2 + 12x = 2x(4x + 6)$.

  Fix Fully factored means the GCF came out. A shared $2$ is still inside, so the answer is $4x(2x + 3)$.

• Trap Adding fractions by combining the denominators: $\dfrac{3}{x} + \dfrac{1}{2} = \dfrac{4}{2x}$.

  Fix Numerators combine only over a shared denominator: over the LCD $2x$, $\dfrac{6}{2x} + \dfrac{x}{2x} = \dfrac{6 + x}{2x}$.

• Trap Letting a subtraction reach only the first term of a numerator: $\dfrac{4x + 1}{x} - \dfrac{x - 2}{x} = \dfrac{3x - 1}{x}$.

  Fix The minus is a factor $-1$ over the whole numerator, so bracket it first: $\dfrac{(4x + 1) - (x - 2)}{x} = \dfrac{4x + 1 - x + 2}{x} = \dfrac{3x + 3}{x}$.