4.3 Equivalent Expressions: Polynomials, Radical & Rational Expressions

4.3 Equivalent Expressions: Polynomials, Radical & Rational Expressions

Skill 3 names three expression types explicitly: polynomials, radical expressions, and rational expressions. The unifying idea is equivalence — rewriting an expression into a different form that has the same value for every permissible input.

Vocabulary and like terms

A term is a product of a coefficient and variable factors. A polynomial is a sum of terms with whole-number exponents. The degree is the largest exponent; the leading coefficient belongs to the highest-degree term.

Like terms have identical variable parts, including exponents:

• 5x² and −8x² are like terms → combine to −3x²
• 5x² and 5x³ are not like terms
• 4xy and 4yx are like terms, since multiplication is commutative
• 6x²y and 6xy² are not like terms — the exponents attach to different variables

Only like terms combine. This single rule blocks the most common invalid step in student work, "3x + 4 = 7x."

Multiplying polynomials

Distribute every term of the first factor across every term of the second.

(2x + 5)(3x − 4) = 6x² − 8x + 15x − 20 = 6x² + 7x − 20

For a binomial times a trinomial, six products appear:

(x + 3)(x² − 2x + 5) = x³ − 2x² + 5x + 3x² − 6x + 15 = x³ + x² − x + 15

The special products

+---------------------------------------------------------------------------+
|  Square of a sum        (a + b)^2 = a^2 + 2ab + b^2                       |
|  Square of a difference (a - b)^2 = a^2 - 2ab + b^2                       |
|  Difference of squares  (a + b)(a - b) = a^2 - b^2                        |
+---------------------------------------------------------------------------+

[!WARNING] (a + b)² ≠ a² + b². The middle term 2ab is not optional. Test it numerically: (3 + 4)² = 49 while 3² + 4² = 25. This error — sometimes called the "freshman's dream" — is the single most frequently cited middle-grades algebra misconception, and Competency 5 items ask you to diagnose it.

Note the asymmetry: the difference of squares factors, but a² + b² does not factor over the real numbers. x² − 16 = (x + 4)(x − 4); x² + 16 is irreducible.

Factoring patterns

Factoring is the distributive property applied in reverse. Work through these in order.

1. Greatest common factor, always first.

12x³ − 18x² = 6x²(2x − 3)

2. Difference of squares.

49y² − 25 = (7y + 5)(7y − 5) 3x² − 27 = 3(x² − 9) = 3(x + 3)(x − 3) — GCF first, then the pattern

3. Trinomials with leading coefficient 1. Find two numbers that multiply to c and add to b.

x² + 7x + 12: 3 and 4 → (x + 3)(x + 4) x² − 5x − 24: 3 and −8 → (x + 3)(x − 8)

4. Trinomials with a leading coefficient. Find two numbers multiplying to a · c and adding to b, then split the middle term.

6x² + 11x − 10: a · c = −60, and 15 · (−4) = −60 with 15 + (−4) = 11. 6x² + 15x − 4x − 10 = 3x(2x + 5) − 2(2x + 5) = (2x + 5)(3x − 2)

5. Perfect square trinomials.

x² + 10x + 25 = (x + 5)² — recognize when the constant is the square of half the linear coefficient

Dividing by a monomial

Division by a single term distributes across every term of the numerator, which is the reverse of the GCF step above.

(12x³ − 18x² + 6x)/(6x) = 2x² − 3x + 1

The vanishing term is the trap: 6x ÷ 6x is 1, not 0. Watch the sign on every term as well, since a negative divisor flips them all.

Checking equivalence by substitution

Because equivalent expressions agree for every permissible input, one substitution can eliminate wrong answer choices quickly. Pick a value that is easy but not degenerate — 2 or 3 rather than 0 or 1, which make too many candidates agree.

Is (x + 3)² equal to x² + 9? Substituting x = 2 gives 25 versus 13, so no. Is 6x² + 11x − 10 equal to (2x + 5)(3x − 2)? At x = 2, 24 + 22 − 10 = 36 and (9)(4) = 36, which supports equivalence.

Substitution can disprove equivalence outright with one counterexample; a single match only supports it, so confirm the algebra when two choices survive. The same method is the fastest way to test a student's claimed rewrite in Competency 5 items.

Rational expressions

A rational expression is a ratio of polynomials. Simplify by factoring both parts completely and cancelling common factors.

(x² − 9)/(x² + x − 12) = [(x + 3)(x − 3)]/[(x + 4)(x − 3)] = (x + 3)/(x + 4), x ≠ 3, x ≠ −4

[!WARNING] You may cancel factors, never terms. In (x + 5)/5, the 5s cannot cancel, because 5 is a term of the numerator rather than a factor of the whole numerator. Test numerically with x = 5: (5 + 5)/5 = 2, whereas "cancelling" would give x = 5.

Operations follow fraction rules:

• Multiply: factor, cancel across the product, multiply what remains.
• Divide: multiply by the reciprocal.
• Add or subtract: find the least common denominator by factoring every denominator.

3/(x + 2) + 5/(x − 1) = [3(x − 1) + 5(x + 2)]/[(x + 2)(x − 1)] = (8x + 7)/[(x + 2)(x − 1)]

Always state excluded values — the inputs that make any original denominator zero. They remain excluded even after cancellation removes them from view.

Radical expressions with variables

The rules from numerical radicals carry over, with one caution about even roots and signs.

• √(x⁸) = x⁴, and √(50x⁵) = √(25x⁴ · 2x) = 5x²√(2x)
• (x^(2/3))(x^(1/2)) = x^(2/3 + 1/2) = x^(7/6)
• (8x⁶)^(1/3) = 2x²

For even roots, √(x²) = |x| in full generality, since the principal square root is nonnegative. Middle-grades items usually restrict variables to positive values, but the absolute value convention is what makes the identity true for all real x.