Polynomials, Exponentials, and Logarithms: Chapter Review

Vocabulary and notation

Polynomial

A finite sum of terms, each a real coefficient times a nonnegative-integer power of $x$. No variable in a denominator or under a root, and no negative or fractional exponent.

Term and coefficient

A term is one coefficient times one power of $x$; the coefficient is its numerical factor with its sign, as $-5$ is in $-5x^2$.

Degree

Of a term, its power of $x$ ($7x$ is degree $1$, a constant degree $0$); of a polynomial, the largest degree among its terms.

Leading term, leading coefficient, constant term

The highest-degree term, its coefficient (never $0$), and the $x^0$ term: in $6x^3 - x^5 + 4$, $-x^5$, $-1$, and $4$.

Standard form

Terms in descending order of degree, highest power first and the constant last.

Monomial, binomial, trinomial

Polynomials of one, two, and three terms.

Like terms

Terms carrying the same power of $x$. Only these merge into one term.

Closure

Adding, subtracting, or multiplying polynomials always gives another polynomial.

Exponential function

The variable in the exponent over a fixed base $b > 0$ with $b \ne 1$: $f(x) = a \cdot b^{x}$.

Power function

Variable in the base under a fixed exponent, as in $x^2$. The opposite arrangement, not exponential.

Principal $P$, rate $r$, frequency $n$

The amount deposited, the yearly rate as a decimal ($5\%$ is $r = 0.05$), and the compounding periods per year.

The number $e$

The irrational constant $e \approx 2.71828$ that $\left(1 + \tfrac{1}{m}\right)^{m}$ climbs toward as $m$ grows.

Logarithm

An exponent. Read $\log_{b}(y)$ as "the exponent that turns $b$ into $y$": $b^{x} = y$ is exponential form, $\log_{b}(y) = x$ logarithmic form.

Common log and natural log

$\log(y)$ with no base written means $\log_{10}(y)$; $\ln(y)$ means $\log_{e}(y)$.

Formulas and theorems

• Combining like terms

  $$c_1 x^{k} + c_2 x^{k} = (c_1 + c_2)x^{k}$$

  Use when The same power $k$ on both. Add the coefficients and keep the power; different powers never merge.

• Subtracting a polynomial

  $$-(A + B + C) = -A - B - C$$

  Use when A factor of $-1$ on EVERY term inside, however many there are.

• Degree of a sum or difference

  $$\deg(p \pm q) \le \max(\deg p, \deg q)$$

  Use when Equality unless $p$ and $q$ share a degree AND their leading coefficients cancel, the only way a degree can drop; then it falls to the next surviving power, or the result is $0$, whose degree is left undefined.

  e.g. $(5x^3 + x) - (5x^3 - 2) = x + 2$: degree $1$ from two degree-$3$ polynomials.

• One term times one term

  $$(a x^{m})(c x^{n}) = ac\,x^{m+n}$$

  Use when Multiply the coefficients, ADD the exponents. A negative sign folds into the coefficient.

  e.g. $(-4x^5)(3x^2) = -12x^7$.

• Two binomials (FOIL)

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

  Use when Four products (First, Outer, Inner, Last), and only for two terms times two terms. The rule that always works: every term of the first times every term of the second, $mn$ products in all. A lone term distributes the same way.

  e.g. $(x - 5)(x + 2) = x^2 - 3x - 10$.

• Degree of a product

  $$\deg(pq) = \deg p + \deg q$$

  Use when Neither factor the zero polynomial. No exception: leading terms multiply, and nonzero times nonzero cannot cancel.

• Exponential $y = a\,b^{x}$, and its graph

  $$y = a\,b^{x}$$

  Text description

  Two exponential curves meeting at the point (0, a): the growth curve flattens toward the asymptote y = 0 on the left and climbs on the right, and the decay curve is its mirror image.

  Use when $a > 0$, $b > 0$, $b \ne 1$: a negative base breaks at fractional exponents, $b = 1$ is a constant. Growth when $b > 1$, decay when $0 < b < 1$. Through $(0, a)$, horizontal asymptote $y = 0$ (flat side left for growth, right for decay), domain all real numbers, range $y > 0$, one-to-one, which is why $b^{x}$ has an inverse, the logarithm.

  e.g. $y = 3 \cdot 2^{x}$ grows from $3$; $y = 5\left(\tfrac{1}{3}\right)^{x}$ decays from $5$.

• Adding inputs multiplies outputs

  $$b^{x + y} = b^{x} \cdot b^{y}$$

  Use when $b > 0$, any real $x$ and $y$. With $y = 1$: each unit step right multiplies the output by $b$, the signature of an exponential.

  e.g. $3^{2} \cdot 3^{4} = 3^{6} = 729$.

• Simple interest

  $$A = P(1 + rt)$$

  Use when $r$ a decimal, $t$ in years. Linear: the same $Pr$ is added every year, never interest on interest.

  e.g. $1000$ dollars at $5\%$ for $3$ years: $1000(1 + 0.05 \cdot 3) = 1150$ dollars.

• Compound interest

  $$\begin{gathered} A = P\left(1 + \frac{r}{n}\right)^{nt} \\ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \end{gathered}$$

  Use when $n$ is compounding periods per YEAR ($2$ semiannual, $4$ quarterly, $12$ monthly), $r$ a decimal, $t$ in years. The rate is divided by $n$ AND the exponent multiplied by $n$. Annual is $n = 1$: $A = P(1 + r)^{t}$. Solving for $P$ needs no logarithm: the unknown is a plain factor, so division finishes it.

  e.g. $1500$ dollars at $6\%$ semiannually for $2$ years: $1500(1.03)^{4} = 1688.26$ dollars.

• Continuous compounding

  $$A = P e^{rt}$$

  Use when The limit as $n$ grows without bound, and a CEILING: with $r > 0$, compounding more often earns a little more each time, by shrinking amounts, but no finite schedule reaches $Pe^{rt}$.

  e.g. $2000$ dollars at $3\%$ for $1$ year: $2000e^{0.03} = 2060.91$ dollars.

• Definition of the logarithm

  $$\log_{b}(y) = x \iff b^{x} = y$$

  Use when Base $b > 0$ with $b \ne 1$, input $y > 0$; $\log_{b}(0)$ and the log of a negative are undefined, a positive base to a real power being always positive. Read off $\log_{b}(1) = 0$, $\log_{b}(b) = 1$.

• The logarithm and the exponential undo each other

  $$b^{\log_{b}(y)} = y, \qquad \log_{b}(b^{x}) = x$$

  Use when $b > 0$ with $b \ne 1$. The first needs $y > 0$; the second holds for every real $x$.

• The three laws of logarithms

  $$\begin{gathered} \log_{b}(MN) = \log_{b} M + \log_{b} N \\ \log_{b}\!\left(\frac{M}{N}\right) = \log_{b} M - \log_{b} N \\ \log_{b}(M^{k}) = k\log_{b} M \end{gathered}$$

  Use when One base throughout, $b > 0$, $b \ne 1$, $M > 0$, $N > 0$, $k$ any real. Products become sums, quotients differences, powers coefficients. The power law moves an exponent on the INPUT out front, so $\log_{b}(M^{k})$ is not $\left(\log_{b} M\right)^{k}$. There is NO law for the log of a sum.

• Change of base, and solving for an exponent

  $$\log_{b}(y) = \frac{\log(y)}{\log(b)}$$

  Use when $y > 0$, $b > 0$, $b \ne 1$. One base top and bottom, $\ln$ as readily as $\log$. The answer is a QUOTIENT.

  e.g. $\log_{2}(10) = \frac{\log 10}{\log 2} = \frac{1}{0.3010} \approx 3.32$.

• Graph of $y = \log_{b}(x)$

  The reflection of $y = b^{x}$ across $y = x$: through $(1, 0)$ and $(b, 1)$, vertical asymptote $x = 0$, domain $x > 0$, range all real numbers.

  Text description

  The exponential and the logarithm are mirror images across the line y = x, so their points, intercepts, and asymptotes all swap coordinates.

  Use when $b > 0$ with $b \ne 1$; increasing when $b > 1$, decreasing when $0 < b < 1$. Domain and range are the exponential's, swapped.

Problem types, step by step

Exam traps

• Trap Splitting the logarithm of a SUM: $\log_{2}(4 + 4) = \log_{2}(4) + \log_{2}(4) = 4$.

  Fix No law breaks up the log of a sum. Add inside first: $\log_{2}(4 + 4) = \log_{2}(8) = 3$. The product law needs a PRODUCT.

• Trap Flipping only the first term of a subtraction: $(5x^2 - 3x + 4) - (2x^2 + 6x - 7) = 3x^2 + 3x - 3$.

  Fix All three signs flip: $5x^2 - 3x + 4 - 2x^2 - 6x + 7 = 3x^2 - 9x + 11$.

• Trap Squaring a sum term by term: $(x + 4)^2 = x^2 + 16$.

  Fix $(x + 4)^2$ is $(x + 4)(x + 4)$, four products with two cross terms: $x^2 + 8x + 16$.

• Trap Doing the wrong thing to exponents: $3x^2 + 5x^2 = 8x^4$ when adding, $x^2 \cdot x^3 = x^6$ when multiplying.

  Fix Adding like terms adds the COEFFICIENTS and leaves the power alone, giving $8x^2$. Multiplying powers ADDS the exponents, giving $x^5$.

• Trap Reading $b^{0}$ as $0$, or a negative exponent as a negative output, so $2^{-3}$ looks like $-8$.

  Fix $b^{0} = 1$ for every allowed base, and a negative exponent gives a reciprocal: $2^{-3} = \tfrac{1}{8}$, positive like every exponential output.

• Trap Treating $b^{x}$ and $x^{b}$ as the same species.

  Fix $2^{x}$ is exponential, $x^{2}$ a power function: at $x = 3$ they give $8$ and $9$, and past $x = 4$ the exponential outruns it for good.

• Trap Putting the percent itself into an interest formula: $A = 1000(1 + 5)^{t}$ for $5\%$.

  Fix The rate is the decimal, so $A = 1000(1.05)^{t}$. Writing $5$ multiplies the balance by $6$ every year.

• Trap Solving $b^{x} = y$ as $x = \log(y) - \log(b)$, or as $x = \log\!\left(\tfrac{y}{b}\right)$.

  Fix Both are the quotient law, a different situation. Dividing $x\log(b) = \log(y)$ by $\log(b)$ leaves $x = \dfrac{\log(y)}{\log(b)}$, a QUOTIENT.