Question

Which polynomial identity will prove that 19 = 27 − 8? Difference of Squares Difference of Cubes Sum of Cubes Square of a Binomial

Answer 1

B

Step-by-step explanation:

difference of cubes

Answer 2

The correct answer is: [B]: "Difference of Cubes".
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Step-by-step explanation:
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Note that the equation/identity for the "difference of cubes" is expressed as:
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" a³ − b³ = (a − b)(a² + ab + b²) " ;
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Note the given equation: " 19 = 27 − 8 " ; → (which is true).
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The "right hand side" of this equation:

→ " 27 − 8 " ; contains two numbers:

→ "27" and "8" ; both of which are "cubes" ;

→ that is: ∛27 = 3 ; ↔ 3³ = 3 * 3 * 3 = 9 * 3 = 27 ; and:

∛ 8 = 2 ; ↔ 2³ = 2 * 2 * 2 = 4 * 2 = 8 ;

→ AND: "8" is being SUBTRACTED from "27" ;

→ (hence, the "difference of squares" polynomial identity);

So: given: " 19 = 27 − 8 " ;

→ Rewrite as:

" 19 = 3³ − 2³ " ;
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Now, consider the identity equation for the "difference of squares":

→ " a³ − b³ = (a − b)(a² + ab + b²) " ;
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Take: " 19 = 3³ − 2³ " ;

and rewrite as:

→ 3³ − 2³ = 19 ;

So: (a³ − b³) = 3³ − 2³ ;

a = 3 ; b = 2 ;
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Plug in these values:

" a³ − b³ = (a − b)(a² + ab + b²) " ;

→ 3³ − 2³ ≟ [3 − 2) [ 3² + (3*2) + 2² ] ≟ 19 ? ;

→ 27 − 8 ≟ (1) (9 + 6 + 4) ≟ 19 ? ;

→ 19 ≟ (1) (15 + 4) ≟ 19 ? ;

→ 19 ≟ (1) (19) ≟ 19 ? ;

→ 19 ≟ 19 ≟ 19 ? ;

→ 19 = 19 = 19 ! Yes!
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