Question

which polynomial can be simplified to a difference of squares?
10a² + 3a − 3a − 16
16a² − 4a + 4a − 1
25a² + 6a − 6a + 36
24a² − 9a + 9a + 4

Explanation:

Step1: Recall difference of squares

The difference of squares formula is \(x^2 - y^2=(x - y)(x + y)\), so the polynomial should simplify to \(A^2 - B^2\), meaning the middle terms (linear terms) should cancel out, and we have a square term minus another square term.

Step2: Simplify each option

• Option 1: \(10a^2+3a - 3a - 16=10a^2-16\). But \(10a^2\) is not a perfect square (since \(10\) is not a perfect square), so this is not a difference of squares.
• Option 2: \(16a^2-4a + 4a - 1=16a^2-1\). Here, \(16a^2=(4a)^2\) and \(1 = 1^2\), so it simplifies to \((4a)^2-1^2\), which is a difference of squares.
• Option 3: \(25a^2+6a - 6a + 36=25a^2 + 36\). This is a sum of squares, not a difference.
• Option 4: \(24a^2-9a + 9a + 4=24a^2+4\). \(24a^2\) is not a perfect square, and it's a sum, not a difference.

Answer:

The second option: \(16a^2 - 4a + 4a - 1\) (which simplifies to \(16a^2-1=(4a)^2 - 1^2\))