➕ Algebra 1 · Factoring

Difference of Squares: true or false

Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.

All statements and answers

• x² − 144 = (x + 12)(x − 12).

  ✓ True. 144 = 12², and (x + 12)(x − 12) = x² − 12x + 12x − 144 = x² − 144.

• m² − 25 = (m − 5)(m − 5).

  ✗ False. (m − 5)(m − 5) = m² − 10m + 25, which has a middle term. The correct factoring has opposite signs: (m + 5)(m − 5). Test with m = 6: m² − 25 = 11, but (6 − 5)(6 − 5) = 1.

• x² + 36 = (x + 6)(x − 6).

  ✗ False. (x + 6)(x − 6) = x² − 36, not x² + 36. Test with x = 1: 1 + 36 = 37, but (1 + 6)(1 − 6) = −35. A sum of squares like this is prime.

• x² + 4y² cannot be factored using integers.

  ✓ True. It is a sum of squares, x² + (2y)². Opposite signs would give a minus sign, and equal signs give a middle term, since (x + 2y)² = x² + 4xy + 4y². No choice of signs works.

• 8x² − 18 = 2(2x + 3)(2x − 3).

  ✓ True. Take out the GCF 2: 2(4x² − 9). Then 4x² − 9 = (2x + 3)(2x − 3). Check at x = 2: 8(4) − 18 = 14 and 2(7)(1) = 14.

• The expression (y² + 25)(y² − 25) is the completely factored form of y⁴ − 625.

  ✗ False. The product is correct, but y² − 25 is a difference of squares and factors again as (y + 5)(y − 5). The complete answer is (y² + 25)(y + 5)(y − 5).

• x⁵ − 9 is not a difference of squares.

  ✓ True. The exponent 5 is odd, so x⁵ is not the square of any power of x (squaring doubles the exponent). So x⁵ − 9 does not have the shape of a difference of squares.

• (6x + 5)(6x − 5) = 36x² − 25.

  ✓ True. Square the first piece, square the second and subtract: (6x)² − 5². The middle terms −30x and +30x cancel.

• 4x² − 9y² cannot be factored because it has two variables.

  ✗ False. 4x² = (2x)² and 9y² = (3y)², so 4x² − 9y² = (2x + 3y)(2x − 3y). The middle terms −6xy and +6xy cancel.

• −x² + 49 = (7 + x)(7 − x).

  ✓ True. Reorder the terms: −x² + 49 = 49 − x², a difference of squares with a = 7 and b = x. Check: (7 + x)(7 − x) = 49 − 7x + 7x − x² = 49 − x².

• 104 × 96 = 100² − 4² = 9984.

  ✓ True. 104 = 100 + 4 and 96 = 100 − 4, so the product is 100² − 4² = 10000 − 16 = 9984.

• 50x² − 8 cannot be factored, because neither 50 nor 8 is a perfect square.

  ✗ False. The terms share the GCF 2: 50x² − 8 = 2(25x² − 4) = 2(5x + 2)(5x − 2). Look for a GCF before you decide.

• x² + 100 cannot be factored even if you allow any real numbers.

  ✓ True. Suppose x² + 100 = (x + p)(x + q). Putting x = −p makes the right side 0, but the left side is p² + 100, which is at least 100. So no real numbers work.

• a² − b² = (a − b)(a − b).

  ✗ False. (a − b)(a − b) = a² − 2ab + b². The correct factoring uses opposite signs: (a + b)(a − b). Test with a = 5 and b = 3: 25 − 9 = 16, but (5 − 3)(5 − 3) = 4.