3.2 Exponents & Polynomials

The Exponent Rules

An exponent tells how many times a base is multiplied by itself: x⁴ = x·x·x·x. A small set of rules governs all exponent arithmetic, and ALEKS expects fluency with each.

Worked example: Simplify (3x²y)(4x³y⁵).

• Multiply coefficients: 3·4 = 12.
• Add exponents on x: x²·x³ = x⁵. Add exponents on y: y¹·y⁵ = y⁶.
• Result: 12x⁵y⁶.

Worked example: Simplify (2x⁴)³.

• Apply the power-of-a-product rule: 2³ · (x⁴)³ = 8 · x¹² = 8x¹².

Worked example: Simplify (15x⁶y²) / (5x²y⁵).

• Coefficients: 15/5 = 3. Subtract exponents: x⁶⁻² = x⁴ and y²⁻⁵ = y⁻³.
• Rewrite the negative exponent: 3x⁴ / y³.

Scientific Notation

Scientific notation expresses a number as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. A positive exponent shifts the decimal right (large numbers); a negative exponent shifts it left (small numbers).

Worked example: Write 47,300 in scientific notation. Move the decimal 4 places left to get 4.73, so 47,300 = 4.73 × 10⁴.

Worked example: Write 0.00086 in scientific notation. Move the decimal 4 places right to get 8.6, so 0.00086 = 8.6 × 10⁻⁴.

Worked example (multiplying): (3 × 10⁵)(2 × 10⁻²) = (3·2) × 10⁵⁺⁽⁻²⁾ = 6 × 10³.

Polynomial Vocabulary

A polynomial is a sum of terms, each a coefficient times a variable raised to a whole-number power. A one-term polynomial is a monomial, two terms a binomial, three a trinomial. The degree is the highest exponent. To add or subtract polynomials, combine like terms (same variable, same exponent); subtraction requires distributing the minus sign to every term.

Worked example (subtraction): (4x² − 3x + 5) − (x² + 2x − 1).

• Distribute the minus: 4x² − 3x + 5 − x² − 2x + 1.
• Combine like terms: (4x² − x²) + (−3x − 2x) + (5 + 1) = 3x² − 5x + 6.

Multiplying Polynomials and FOIL

To multiply a monomial by a polynomial, distribute: 2x(3x − 4) = 6x² − 8x. To multiply two binomials, use FOIL — multiply the First terms, Outer terms, Inner terms, and Last terms, then combine.

Worked example: (x + 3)(x − 5).

• First: x·x = x²
• Outer: x·(−5) = −5x
• Inner: 3·x = 3x
• Last: 3·(−5) = −15
• Combine the middle terms: x² − 5x + 3x − 15 = x² − 2x − 15.

Special Products

Three patterns appear constantly and are worth memorizing so you can skip FOIL:

• Difference of squares: (a + b)(a − b) = a² − b². Example: (x + 7)(x − 7) = x² − 49.
• Square of a sum: (a + b)² = a² + 2ab + b². Example: (x + 4)² = x² + 8x + 16.
• Square of a difference: (a − b)² = a² − 2ab + b². Example: (x − 3)² = x² − 6x + 9.

Common Mistakes

• Writing (a + b)² = a² + b² — WRONG; you must include the middle term 2ab.
• Adding exponents when you should multiply: (x²)³ = x⁶, not x⁵.
• Treating x⁰ as 0; any nonzero base to the zero power is 1.
• Forgetting to distribute the subtraction sign to EVERY term of the second polynomial.
• Dropping the negative when converting a small number to scientific notation.

Combining Exponent Rules

Real ALEKS items often combine several rules in one expression. Work from the inside out, applying the power rules before the product or quotient rules.

Worked example: Simplify (2x³)² · x⁴.

• Apply the power rule to the first factor: (2x³)² = 4x⁶.
• Multiply by x⁴ using the product rule: 4x⁶ · x⁴ = 4x¹⁰.

Worked example: Simplify (x⁵ / x²)³.

• Inside first: x⁵/x² = x³.
• Then the outer power: (x³)³ = x⁹.

Worked example (negative exponent in a quotient): Simplify (4x⁻²) / (x³).

• Subtract exponents: x⁻²⁻³ = x⁻⁵.
• Rewrite with a positive exponent: 4 / x⁵.

Dividing in Scientific Notation

To divide numbers in scientific notation, divide the leading numbers and subtract the exponents of 10.

Worked example: (8 × 10⁷) / (2 × 10³) = (8/2) × 10⁷⁻³ = 4 × 10⁴.

If the leading number ends up outside the range 1 ≤ a < 10, adjust it. For instance, 12 × 10⁵ becomes 1.2 × 10⁶ by moving the decimal one place left and increasing the exponent by 1.

Multiplying a Binomial by a Trinomial

FOIL only works for two binomials. For larger products, distribute each term of the first factor across every term of the second, then combine like terms.

Worked example: (x + 2)(x² + 3x + 1).

• Distribute x: x³ + 3x² + x.
• Distribute 2: 2x² + 6x + 2.
• Combine like terms: x³ + 5x² + 7x + 2.