Polynomial Identities (Grades 11-12)

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Polynomial Identities

1. 
Factor. [math]27x^3 + 64[/math]
  1. [math](3x-4)(9x^2 + 12x + 16) [/math]
  2. [math](3x + 4)(9x^2 - 12x + 16)[/math]
  3. [math](3x+4)(9x^2 + 12x + 16)[/math]
  4. [math]"Can't factor further"[/math]
2. 
Which one of the following is NOT true?
  1. [math](a+b)^2=a^2+2ab+b^2[/math]
  2. [math]a^2+4a+4=(a+2)^2[/math]
  3. [math]a^2-2ab+b^2=(a-b)^2[/math]
  4. [math]a^2-6ab-9b^2=(a-3b)^2[/math]
  5. [math](3a+2b)^2=9a^2+12ab+4b^2[/math]
3. 
Which one of the following is NOT true?
  1. [math](x+2)(x-2)=x^2-4[/math]
  2. [math](30+2)(30-2)=900-4[/math]
  3. [math](2x+1)(2x-1)=4x^2-1[/math]
  4. [math](a+b)(a-b)=a^2-b^2[/math]
  5. [math](2a+3b)(2a-3b)=2a^2-3b^2[/math]
4. 
Which of the following is a correct polynomial identity that illustrates the statement "the sum of any integer and its value squared is an even number"?
  1. [math]n^2 + n = 2n[/math]
  2. [math]n^2 + n = n/2[/math]
  3. [math]n^2 + n = n(n+1)[/math]
  4. [math]n^2 + n = n^2(1+n)[/math]
5. 
Sam claims that for any two consecutive integers, the difference of the larger number squared and the smaller number squared is always odd. He writes an identity he says proves this, which is [math](n+1)^2 - n^2 = 2n-1[/math]. Is this true? If not, choose the reason why.
  1. It is true.
  2. [math]2n-1[/math] is not always an odd number.
  3. It is not an identity.
  4. It is only true for some numbers.
6. 
Is the following proof correct? If not, identify the first step that has an error.

[math]"Prove: " \ (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2[/math]

Left Side: [math](x^2+y^2)^2 [/math]
Step 1: [math](x^2+y^2)(x^2+y^2)[/math]
Step 2: [math]x^4-2x^2y^2+4x^2y^2+y^4[/math]
Step 3: [math](x^2-y^2)^2+(2xy)^2[/math]
  1. The proof is correct.
  2. Step 1 has an error.
  3. Step 2 has an error.
  4. Step 3 has an error.
7. 
Is the proof correct? If not, identify the first step that has an error.

[math]"Prove: " \ (a+b)^2 = a^2 + 2ab + b^2[/math]

Left Side: [math](a+b)^2[/math]
Step 1: [math](a+b)(a+b)[/math]
Step 2: [math]a^2+ab+ab+b^2[/math]
Step 3: [math]a^2+2ab+b^2[/math]
  1. The proof is correct.
  2. Step 1 has an error.
  3. Step 2 has an error.
  4. Step 3 has an error.
8. 
Is the following proof correct? If not, identify the first step that has an error.

[math]"Prove: " \ (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2[/math]

Left Side: [math](x^2 + y^2)^2[/math]
Step 1: [math](x^4+y^4)[/math]
Step 2: [math]x^2+2x^2y^2+y^2[/math]
Step 3: [math](x^2-y^2)^2+(2xy)^2[/math]
  1. The proof is correct.
  2. Step 1 has an error.
  3. Step 2 has an error.
  4. Step 3 has an error.
9. 
Is the following proof correct? If not, identify the first step that has an error.

[math]"Prove: " (n+2)^2 - n^2 = 4n+4 [/math]

[math]"Left Side: " (n+2)^2 - n^2 [/math]
[math]"Step 1 " ( \ (n+2) + n \ )( \ (n+2) - 1 \ ) [/math]
[math]"Step 2 " (2n+2) (2) [/math]
[math]"Step 3 " 4n+4 [/math]
  1. The proof is correct.
  2. Step 1 has an error.
  3. Step 2 has an error.
  4. Step 3 has an error.
10. 
Is the following proof correct? If not, identify the first step that has an error.

[math]"Prove: " x^4 - y^4 = (x-y)(x+y)(x^2+y^2)[/math]

[math]"Right Side: " (x-y)(x+y)(x^2+y^2)[/math]
[math]"Step 1: " (x-y)(x^3 + xy^2 + x^2y + y^3)[/math]
[math]"Step 2: " (x-y)(x^3 + y^3)[/math]
[math]"Step 4: " x^4 - y^4[/math]
  1. The proof is correct.
  2. Step 1 has an error.
  3. Step 2 has an error.
  4. Step 3 has an error.

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