Factor completely: x² - 16

The expression x² - 16 factors completely into (x - 4)(x + 4) when recognized as a difference of squares a² - b² = (a - b)(a + b) with a = x and b = 4, producing the product of conjugate binomials that expands back to the original expression without further factorization possible over the real numbers.

A. (x - 4)²

This factorization expands to x² - 8x + 16, which differs from the original x² - 16 by the presence of a linear term -8x and sign change on the constant term. The error likely originates from confusing difference of squares (a² - b²) with perfect square trinomial (a - b)² = a² - 2ab + b² a critical algebraic misconception where learners recognize factoring is required but misapply the specific pattern based on sign configuration and term presence. Students might execute √x² = x and √16 = 4 correctly but then assume identical signs in factors due to the minus sign in the original expression, failing to recognize difference of squares requires opposite signs in the binomial factors to produce cancellation of the cross terms during expansion. Another plausible pathway involves solving x² - 16 = 0 to get x = ±4 then incorrectly constructing factors as (x - 4)(x - 4) through sign duplication error. The presence of the linear term -8x upon expansion immediately disqualifies this factorization revealing how sign errors in factor construction produce mathematically invalid results that fail verification through expansion, a critical deficiency requiring explicit emphasis on expansion verification before accepting any factorization to catch sign errors that compromise algebraic equivalence.

B. (x + 4)²

This expression expands to x² + 8x + 16, differing from the original by both sign of the linear term (+8x versus none) and constant term sign (+16 versus -16). The error likely originates from the same pattern confusion as option A but with positive sign selection, possibly influenced by misconception that squares are always positive leading to positive factor signs. Students might recognize 16 as 4² but then assume both factors require positive signs to produce positive constant term upon multiplication, failing to recognize that (+4)(+4) = +16 whereas the original requires -16, necessitating opposite signs in factors to produce negative constant through (+4)(-4) = -16. Another plausible pathway involves misreading the original expression as x² + 16 (sum of squares, unfactorable over reals) then incorrectly factoring as (x + 4)². The dramatic expansion mismatch (x² + 8x + 16 versus x² - 16) reveals fundamental misunderstanding of how binomial sign combinations affect expansion results a conceptual gap requiring explicit expansion practice showing how (x + a)(x + b) = x² + (a+b)x + ab with sign patterns determining coefficient signs, building robust factor construction protocols preventing sign errors that invalidate factorizations.

C. (x - 4)(x + 4)

This factorization correctly represents the complete factorization through multiple verification pathways demonstrating algebraic manipulation mastery. Difference of squares application: x² - 16 = x² - 4² = (x - 4)(x + 4). Expansion verification: (x - 4)(x + 4) = x² + 4x - 4x - 16 = x² - 16, confirming exact equivalence with original expression. Root verification: solving x² - 16 = 0 yields x = ±4, matching zeros of factors (x - 4) = 0 → x = 4 and (x + 4) = 0 → x = -4. Graphical interpretation: parabola y = x² - 16 crosses x-axis at (4,0) and (-4,0), confirming linear factors correspond to x-intercepts. Irreducibility confirmation: neither factor contains common factors or further factorable structure over real numbers, satisfying "completely factored" requirement. This solution demonstrates comprehensive understanding of factoring techniques including pattern recognition (difference of squares), expansion verification to confirm equivalence, root-factor relationship exploitation, graphical interpretation reinforcing algebraic results, and irreducibility assessment ensuring complete factorization integrated competencies essential for calculus (partial fraction decomposition), differential equations (characteristic equation solving), cryptography (integer factorization analogs), and computer algebra systems where accurate factorization enables efficient computation, equation solving, and structural analysis across increasingly complex polynomial manipulations.

D. (x - 8)(x + 2)

This factorization expands to x² - 6x - 16, differing from the original by the presence of linear term -6x and constant term magnitude preservation with incorrect sign configuration. The error likely originates from factoring the constant term 16 as 8 × 2 and incorrectly assuming these are the binomial constants without verifying their sum equals the linear coefficient's opposite (which is 0 for x² - 16). Students might recognize factors of 16 as 8 and 2 but fail to apply the critical sum condition: for x² + bx + c, factor constants m and n must satisfy m + n = b and mn = c. Here, (-8) + 2 = -6 ≠ 0 while (-8)(2) = -16 satisfies the product condition revealing partial application of factoring criteria where learners check only the product condition (mn = c) while neglecting the sum condition (m + n = b). This "product-only" error represents a pervasive deficiency in quadratic factoring where students identify factor pairs of c but omit verification that their sum equals b, producing mathematically invalid factorizations that appear plausible due to correct constant term reproduction. The presence of the linear term -6x upon expansion immediately signals invalidity for a binomial lacking linear term highlighting the necessity of explicit dual-condition verification protocols (both sum and product) before accepting any factor pair as valid, with expansion verification serving as mandatory final check to catch sum condition violations that compromise algebraic equivalence.

Conclusion

The factorization (x - 4)(x + 4) emerges through rigorous application of the difference of squares pattern recognizing x² - 16 as x² - 4², producing conjugate binomial factors with opposite signs that expand back to the original expression through cross-term cancellation. This problem reinforces critical algebraic competencies essential across mathematical domains: recognizing special factoring patterns (particularly difference of squares with its distinctive minus sign between perfect squares and requirement for opposite signs in factors), executing expansion verification to confirm equivalence before accepting factorizations, understanding the relationship between polynomial roots and linear factors, and assessing irreducibility to ensure complete factorization. Mastery of these integrated skills proves indispensable for calculus (integration via partial fractions), abstract algebra (unique factorization domains), number theory (Fermat's factorization method), and engineering (transfer function pole-zero analysis) where accurate factorization enables equation solving, structural decomposition, and system analysis across contexts requiring polynomial manipulation. The distractors strategically target pervasive misconceptions including pattern confusion (difference of squares versus perfect square trinomials), sign management errors in factor construction, and partial criterion application (product without sum verification) highlighting the necessity of explicit pattern recognition protocols with visual scaffolding contrasting different factoring structures, and mandatory expansion verification before final answer acceptance to prevent factorizations that fail algebraic equivalence checks with potentially severe consequences in equation solving contexts where invalid factorizations produce extraneous or missing solutions.