Examine four examples of computations with polynomials.
Identify the correct example and all the errors:
(a) $$ \begin{aligned} (4x^2-5xy+5y^2)-(x^2-12xy+3y^2) &\stackrel{(1)}= 4x^2-5xy+5y^2-x^2+12xy-3y^2 = \cr &\stackrel{(2)}= (4-1)x^2-(5+12)xy+(5-3)y^2 =\cr &\stackrel{(3)}= 3x^2-17xy+2y^2 \end{aligned} $$ (b) $$ \begin{aligned} (4xy+y-8x)-(2x-7y-8)-(-xy-6y+3) &\stackrel{(1)}= 4xy+y-8x+2x-7y-8+xy-6y+3 = \cr &\stackrel{(2)}= (4+1)xy+(1-7-6)y+(-8+2)x+(-8+3) =\cr &\stackrel{(3)}= 5xy-12y-6x-5 \end{aligned} $$ (c) $$ \begin{aligned} (5x-7y+2)+(14x+12y-3)-(3x+5y-1)&\stackrel{(1)}= 5x-7y+2+14x+12y-3-3x-5y+1 = \cr &\stackrel{(2)}= (5+14-3)x+(-7+12-5)y+(2-3+1) =\cr &\stackrel{(3)}= 16x+y \end{aligned} $$ (d) $$ \begin{aligned} (7x^2-6xy+12y^2)-(5x^2-5xy+7y^2) &\stackrel{(1)}= 7x^2-6xy+12y^2-5x^2+5xy-7y^2 = \cr &\stackrel{(2)}= (7-5)x^2+(-6+5)xy+(12-7)y^2 =\cr &\stackrel{(3)}= 2x^2-xy+5y^2 \end{aligned} $$
Choose the correct answer from the four options below.
- The example (a) is computed correctly.
- In (b), there is a mistake in step (1).
- In (c), there is a mistake in step (2).
- In (d), there is a mistake in step (3).
- The example (b) is computed correctly.
- In (a), there is a mistake in step (2).
- In (c), there is a mistake in step (3).
- In (d), there is a mistake in step (2).
- The example (c) is computed correctly.
- In (a), there is a mistake in step (3).
- In (b), there is a mistake in step (1).
- In (d), there is a mistake in step (1).
- The example (d) is computed correctly.
- In (a), there is a mistake in step (2).
- In (b), there is a mistake in step (1).
- In (c), there is a mistake in step (3).