Polynomials and Radicals

• Add, subtract, and multiply polynomials, and factor out a greatest common factor or a difference of squares
• Connect factors and zeros: if (x − r) is a factor then r is a zero, and a polynomial of degree n has at most n real zeros
• Simplify a rational expression by factoring and canceling common factors (never terms), and name the excluded values
• Solve a rational equation by multiplying through by the common denominator, then reject any answer that makes a denominator zero
• Simplify square roots (√50 = 5√2) and solve a radical equation by isolating the root, squaring, and checking for extraneous solutions

Polynomials: add and subtract by combining like terms; multiply by distributing every term (foil for two binomials), so (x + 3)(x − 5) = x² − 2x − 15. Always look for a greatest common factor first: 6x³ − 9x² = 3x²(2x − 3). A difference of squares factors as a² − b² = (a − b)(a + b), so x² − 49 = (x − 7)(x + 7). To expand a binomial power, use the binomial coefficients from Pascal's triangle (1, 3, 3, 1 for a cube; 1, 4, 6, 4, 1 for a 4th power): (x − 3)⁴ = x⁴ − 12x³ + 54x² − 108x + 81, never just x⁴ + 81. Factors and zeros are two views of the same thing: if (x − r) is a factor of a polynomial, then r is a zero and the graph meets the x-axis at x = r. A polynomial of degree n has at most n real zeros, so a cubic crosses the x-axis at most three times. Given one linear factor of a cubic, divide it out and factor the quadratic that is left: x³ − 2x² − 11x + 12 = (x − 4)(x² + 2x − 3) = (x − 4)(x + 3)(x − 1), so the zeros are 4, −3, and 1. Rational expressions are fractions of polynomials. To simplify, factor the top and bottom completely and cancel common factors: (x² − 9)/(x² + 3x) = (x − 3)(x + 3) / (x(x + 3)) = (x − 3)/x. You may cancel factors only, never terms that are added, and the excluded values come from the original denominator (here x ≠ 0 and x ≠ −3). To add rational expressions, build a common denominator just as with numeric fractions. To solve a rational equation, multiply every term by the common denominator: 3/x + 1/2 = 5/x becomes 6 + x = 10 after multiplying by 2x, so x = 4. Then check that the answer does not make any original denominator zero. Radicals: √(ab) = √a × √b, so √50 = √25 × √2 = 5√2. Only like radicals combine: 3√2 + 5√2 = 8√2, but √2 + √3 stays as is, so simplify each root before adding: √75 + √27 = 5√3 + 3√3 = 8√3. A question that says 'whenever the expression is defined, it is equivalent to' is a simplify-by-factoring question. To rationalize 6/√3, multiply top and bottom by √3 to get 2√3. To solve a radical equation, isolate the root and square both sides: √(x + 3) = x − 3 gives x + 3 = x² − 6x + 9, so x² − 7x + 6 = 0 and x = 1 or x = 6. Squaring can create extraneous solutions, so check both: x = 6 works (3 = 3), but x = 1 gives 2 = −2 and must be rejected. Graph features for hard questions: a zero from a factor raised to an even power only touches the x-axis, while an odd power crosses it. The remainder when p(x) is divided by x − a equals p(a) (the remainder theorem), so p(a) = 0 exactly when x − a is a factor. Nonreal zeros of a polynomial with real coefficients come in conjugate pairs. For a rational function, a factor that cancels from the top and bottom leaves a hole, and a factor left in the denominator gives a vertical asymptote. For the horizontal asymptote, compare degrees: a higher bottom degree gives y = 0, equal degrees give the ratio of the leading coefficients, and a higher top degree gives no horizontal asymptote.

• Factor before you do anything else. Most rational expression questions collapse once the top and bottom are factored.
• Cancel factors, never terms. In (x + 3)/(x + 5) nothing cancels.
• After solving a rational or radical equation, substitute each answer into the original equation. The ACT puts the extraneous value among the choices.
• When the choices are numbers, backsolving a rational or radical equation is often faster than the algebra.