Answer
The Step where Shayla made her mistake is STEP 3
SOLUTION
Problem Statement
The question gives us Shayla's workings of simplifying the following expression:
[tex]x^4-25[/tex]- We are asked to check which step she made her first mistake.
Method
- To solve this problem, we shall perform checks at each step.
- We shall try to aggregate the expression in each step and try to get back the expression in the previous step.
- That is, we shall, for example, try to get back step1 by aggregating step 2 and try to get back step 2 by aggregating step 3 and so on.
- Whichever step we aggregate and does not give us the step that precedes it, is the step where Shayla made her mistake.
Implementation
Step 1
[tex]\begin{gathered} (x^2+5)(x^2-5) \\ \text{Let us try to see if we can get back the question from step 1.} \\ \\ \text{Expand the bracket} \\ x^2(x^2)-5(x^2)+5(x^2)-25 \\ x^4-5x^2+5x^2-25 \\ \\ \therefore(x^2+5)(x^2-5)=x^4-25 \\ \\ \text{THUS STEP 1 IS CORRECT} \end{gathered}[/tex]Step 2:
[tex]\begin{gathered} (x^2+5)(x+\sqrt[]{5})(x-\sqrt[]{5}) \\ \text{Let us try to get the expression in step 1 back.} \\ \\ Expand\text{ the brackets as follows:} \\ (x^2+5)(x(x)-x\sqrt[]{5}+x\sqrt[]{5}-\sqrt[]{5}(\sqrt[]{5})) \\ (x^2+5)(x^2-(\sqrt[]{5})^2) \\ (x^2+5)(x^2-5) \\ \\ \text{ THUS STEP 2 IS CORRECT} \end{gathered}[/tex]Step 3
[tex]\begin{gathered} (x+5i)(x-5i)(x+\sqrt[]{5})(x-\sqrt[]{5}) \\ \text{Let us try to get step 2 from this expression} \\ \\ Expand\text{ the brackets as follows:} \\ (x(x)+x(-5i)+x(5i)+5i(-5i))(x+\sqrt[]{5})(x-\sqrt[]{5}) \\ (x^2-5ix+5ix+25)((x+\sqrt[]{5})(x-\sqrt[]{5}) \\ =(x^2+25)(x+\sqrt[]{5})(x-\sqrt[]{5}) \\ \\ (x^2+25)(x+\sqrt[]{5})(x-\sqrt[]{5})\ne(x^2+5)(x+\sqrt[]{5})(x-\sqrt[]{5}) \\ \\ \text{THUS STEP 3 IS NOT CORRECT} \end{gathered}[/tex]Final Answer
The Step where Shayla made her mistake is STEP 3