2.3 Exponents, Radicals, and Polynomials

Where These Skills Show Up

ACT's math description includes integer and rational exponents under Number & Quantity, and polynomial, radical, and exponential relationships under Algebra. That means these skills can appear as direct simplification, as part of a function question, or inside a word model. The question may look advanced, but the winning move is usually one familiar rule applied cleanly.

This topic is also where calculator confidence can become a problem. A calculator may approximate a radical or expand a product, but ACT often tests whether you know which expressions are equivalent. Exact form matters when choices contain radicals, exponents, or factors.

Exponent Rules You Must Control

For example, x^3 * x^5 = x^8 because the bases match. But x^3 * y^5 cannot be combined into one power. In (2x^3)^2, the exponent applies to 2 and to x^3, so the result is 4x^6.

Negative exponents are relocation instructions. 5x^-2 equals 5/x^2, not -5x^2. If a fraction has x^-3 in the denominator, moving it to the numerator gives x^3. This is a common ACT trap because the wrong sign often appears as an answer choice.

Rational exponents connect powers and roots. 27^(2/3) means take the cube root of 27, then square: 3^2 = 9. You could square first and then take the cube root, but root first usually keeps numbers smaller.

Radicals and Restrictions

A radical such as sqrt(50) can simplify because 50 = 25 * 2, so sqrt(50) = 5sqrt(2). The goal is to pull out perfect-square factors. For cube roots, pull out perfect cubes: cubert(54) = cubert(27 * 2) = 3cubert(2).

When radicals contain variables, restrictions matter. In real-number ACT questions, sqrt(x - 3) requires x - 3 >= 0, so x >= 3. A denominator such as 1/(x - 3) requires x != 3. If both restrictions appear, both must be honored.

Radical equations need a final check. Suppose sqrt(x + 6) = x. Squaring gives x + 6 = x^2, so x^2 - x - 6 = 0 and (x - 3)(x + 2) = 0. The candidates are x = 3 and x = -2. In the original equation, sqrt(4) = -2 is false, so x = -2 is extraneous. Only x = 3 works.

Polynomial Operations

A polynomial is a sum of terms with nonnegative integer exponents, such as 3x^2 - 5x + 8. To add or subtract polynomials, combine like terms only. To multiply, distribute every term. FOIL is just distribution for two binomials.

Example: (x - 4)(x + 7) = x^2 + 7x - 4x - 28 = x^2 + 3x - 28. Sign control is the main issue. If the expression is (x - 4)^2, do not square term by term. It equals (x - 4)(x - 4) = x^2 - 8x + 16.

Factoring and Quadratics

Factored form shows zeros. If x^2 - 9x + 20 = 0, find two numbers that multiply to 20 and add to -9: -4 and -5. The factorization is (x - 4)(x - 5) = 0, so x = 4 or x = 5.

The difference of squares pattern is a^2 - b^2 = (a - b)(a + b). Thus x^2 - 49 = (x - 7)(x + 7). The pattern does not work for x^2 + 49 over the real numbers.

When factoring is slow, use the quadratic formula for ax^2 + bx + c = 0: x = (-b +/- sqrt(b^2 - 4ac))/(2a). The discriminant b^2 - 4ac tells the number of real solutions: positive gives two, zero gives one repeated solution, and negative gives none.

Choosing the Right Form

• Use expanded form when combining like terms or reading the y-intercept of a quadratic.
• Use factored form when the question asks for zeros, roots, or x-intercepts.
• Use vertex form for maximum, minimum, or turning point questions.
• Use exponent rules before substituting large numbers.
• Use restrictions before accepting a radical or rational-expression solution.

ACT questions often hide a simple factor behind a longer expression. If every term has a common factor, remove it first. For 6x^3 - 9x^2, factor out 3x^2 to get 3x^2(2x - 3). That form reveals both the common structure and any zero values.

Polynomial evaluation is another place to choose the short route. If f(x) = x^3 - 2x^2 + 5 and the question asks for f(-1), substitute -1 before expanding any related expression: -1 - 2 + 5 = 2. Parentheses around negative inputs are not optional; (-1)^2 is positive, while -1^2 is interpreted as the negative of 1 squared.

Complex Numbers and Powers of i

Number & Quantity can include complex numbers. The key fact is i^2 = -1. Powers of i repeat in a cycle: i, -1, -i, 1, then back to i. For i^23, divide 23 by 4 and use the remainder 3, so i^23 = i^3 = -i.

For products, use i^2 = -1 after multiplying. (3 + 2i)(1 - i) = 3 - 3i + 2i - 2i^2 = 3 - i + 2 = 5 - i. The final step changes -2i^2 into +2 because i^2 equals -1.

The final ACT habit is to check form. If the choices are factored, do not expand unless needed. If the choices are radicals, simplify exactly. If the choices are decimals, estimate whether the calculator result is reasonable before selecting it.