Population counts, computer storage, the distance to a star: huge and tiny numbers are everywhere, and exponents let you write them in a flash. In this chapter you will master the rules of exponents, learn to write numbers in scientific notation, and then move on to polynomials: how to name them, add them, subtract them and multiply them quickly.
1. Product and quotient rules
In \(a^n\), the base \(a\) is multiplied by itself \(n\) times. For example \(x^4 = x\cdot x\cdot x\cdot x\). The small number \(n\) is the exponent.
For any nonzero base \(a\) and whole numbers \(m\) and \(n\):
\[ a^m \cdot a^n = a^{m+n} \qquad\qquad \dfrac{a^m}{a^n} = a^{m-n} \]
Same base: you add the exponents when multiplying and subtract them when dividing.
Why does it work? \(x^3\cdot x^2 = (x\cdot x\cdot x)(x\cdot x) = x^5\): five factors of \(x\) in total.
Simplify \(3x^2\cdot 4x^6\) and \(\dfrac{15y^9}{3y^4}\).
Multiply the coefficients, add the exponents: \(3x^2\cdot 4x^6 = 12x^{2+6} = 12x^8\).
Divide the coefficients, subtract the exponents: \(\dfrac{15y^9}{3y^4} = 5y^{9-4} = 5y^5\).
The rule needs the same base. \(x^3\cdot y^2\) cannot be simplified, and \(2^3\cdot 2^4 = 2^7\), not \(4^7\): the bases are never multiplied.
2. Power rule and zero exponent
\[ (a^m)^n = a^{mn} \qquad (ab)^n = a^n b^n \qquad \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \]
A power of a power: multiply the exponents. A power of a product: every factor gets the exponent.
For every nonzero number \(a\), \(a^0 = 1\).
Look at the pattern: \(\dfrac{x^5}{x^5} = 1\) because anything divided by itself is 1, and the quotient rule gives \(x^{5-5} = x^0\). So \(x^0\) must equal 1.
Simplify \((2x^3)^4\) and \((-3a^2b)^3\).
\((2x^3)^4 = 2^4\cdot (x^3)^4 = 16x^{12}\).
\((-3a^2b)^3 = (-3)^3\cdot a^{6}\cdot b^3 = -27a^6b^3\).
\((2x)^3 = 8x^3\), not \(2x^3\): the exponent applies to the 2 as well.
3. Negative exponents
For \(a \neq 0\) and a positive whole number \(n\): \[ a^{-n} = \dfrac{1}{a^n} \]
A negative exponent does not make the number negative; it flips the base to the other side of the fraction bar.
| n | 3 | 2 | 1 | 0 | -1 | -2 | -3 |
|---|---|---|---|---|---|---|---|
| \(2^n\) | 8 | 4 | 2 | 1 | \(\frac12\) | \(\frac14\) | \(\frac18\) |
In the table, each step to the right divides by 2. Continuing past \(2^0 = 1\) leads naturally to \(\frac12\), \(\frac14\), \(\frac18\).
Rewrite with positive exponents: \(5^{-2}\), \(\dfrac{x^3}{x^7}\), and \(4x^{-2}y^3\).
\(5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}\).
\(\dfrac{x^3}{x^7} = x^{3-7} = x^{-4} = \dfrac{1}{x^4}\).
\(4x^{-2}y^3 = \dfrac{4y^3}{x^2}\): only \(x\) carries the negative exponent, so only \(x\) moves.
4. Scientific notation
A number is in scientific notation when it is written \[ a \times 10^n \quad\text{with } 1 \le a < 10 \text{ and } n \text{ an integer.} \]
- Place the decimal point after the first nonzero digit to get \(a\).
- Count how many places the point moved: that is \(|n|\).
- Large number: \(n\) is positive. Number between 0 and 1: \(n\) is negative.
Write 45,000,000 and 0.00032 in scientific notation.
\(45{,}000{,}000 = 4.5\times 10^{7}\) (the point moved 7 places left).
\(0.00032 = 3.2\times 10^{-4}\) (the point moved 4 places right).
To multiply or divide, handle the decimals and the powers of 10 separately, then fix the result if \(a\) leaves the interval from 1 to 10.
Compute \((8\times 10^4)(5\times 10^3)\) and \(\dfrac{6\times 10^8}{1.5\times 10^3}\).
\((8\times 10^4)(5\times 10^3) = 40\times 10^{7} = 4\times 10^{8}\).
\(\dfrac{6\times 10^8}{1.5\times 10^3} = 4\times 10^{5}\).
5. Classifying polynomials
A monomial is a number, a variable, or a product of a number and variables with whole-number exponents, such as \(7x\) or \(-4x^3\). A polynomial is a monomial or a sum of monomials, called its terms.
- The degree of a term is its exponent on the variable (a plain number has degree 0).
- The degree of a polynomial is the largest degree of its terms.
- In standard form the terms go from highest to lowest degree; the first coefficient is the leading coefficient.
- One term: monomial. Two terms: binomial. Three terms: trinomial.
Describe \(P(x) = 4x - 7 + 2x^3\).
Standard form: \(2x^3 + 4x - 7\). It has three terms, so it is a trinomial. Its degree is 3 and its leading coefficient is 2.
The graph above belongs to a degree 2 polynomial. A degree 2 polynomial draws a parabola, and the points where it meets the x-axis are the values that make the polynomial equal to 0.
6. Adding and subtracting polynomials
- To subtract, change the sign of every term of the second polynomial.
- Group the like terms (same variable, same exponent).
- Add the coefficients; the exponents do not change.
Compute \((3x^2+5x-4)+(2x^2-7x+9)\) and \((6x^2-x+3)-(2x^2+4x-5)\).
Sum: \(5x^2 - 2x + 5\).
Difference: \(6x^2 - x + 3 - 2x^2 - 4x + 5 = 4x^2 - 5x + 8\).
In \(A - (x - 5)\), the sign of both terms flips: \(-x + 5\), not \(-x - 5\).
7. Multiplying polynomials
Multiplying uses the distributive property, and each product of terms follows the product rule for exponents.
The rectangle has sides \(x+3\) and \(x+2\). Its total area is the sum of four pieces: \((x+3)(x+2) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6\). The pattern is called FOIL (First, Outer, Inner, Last) for two binomials.
Expand \(3x(x^2-4x+2)\) and \((2x-3)(x^2+4x-1)\).
\(3x(x^2-4x+2) = 3x^3 - 12x^2 + 6x\).
\((2x-3)(x^2+4x-1) = 2x^3 + 8x^2 - 2x - 3x^2 - 12x + 3 = 2x^3 + 5x^2 - 14x + 3\).
Check at \(x = 1\): \((-1)(4) = -4\) and \(2+5-14+3 = -4\).
On my planet we test every expansion by plugging in a small number such as \(x = 1\) or \(x = 2\) in both the factored and the expanded form. If the two results differ, hunt for the error!
8. Special products
\[ (a+b)^2 = a^2 + 2ab + b^2 \]
\[ (a-b)^2 = a^2 - 2ab + b^2 \]
\[ (a+b)(a-b) = a^2 - b^2 \]
The square of side \(a+b\) is cut into \(a^2\), two rectangles \(ab\) and \(b^2\). Adding them gives the first identity. In the last one, the middle terms \(+ab\) and \(-ab\) cancel.
\((x+6)^2 \neq x^2 + 36\). Do not forget the middle term \(2ab\), here \(12x\).
Expand \((x+6)^2\), \((3x-5)^2\), \((4x+7)(4x-7)\); then compute \(51^2\) mentally.
\((x+6)^2 = x^2 + 12x + 36\).
\((3x-5)^2 = 9x^2 - 30x + 25\).
\((4x+7)(4x-7) = 16x^2 - 49\).
\(51^2 = (50+1)^2 = 2500 + 100 + 1 = 2601\).
Key takeaways
- Same base: \(a^m a^n = a^{m+n}\), \(\dfrac{a^m}{a^n} = a^{m-n}\), \((a^m)^n = a^{mn}\).
- \(a^0 = 1\) for \(a \neq 0\), and \(a^{-n} = \dfrac{1}{a^n}\).
- Scientific notation: \(a\times 10^n\) with \(1 \le a < 10\).
- A polynomial is named by its number of terms and its degree; write it in standard form.
- To add or subtract, combine like terms; flip every sign when subtracting.
- To multiply, distribute every term of one factor to every term of the other.
- \((a\pm b)^2 = a^2 \pm 2ab + b^2\) and \((a+b)(a-b) = a^2 - b^2\).
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