➕ Algebra 1 · Exponents and Polynomials
Special Products of Binomials: true or false
Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.
All statements and answers
(n + 12)2 = n2 + 24n + 144.
✓ True. Double the product: 2 × n × 12 = 24n. Square the last term: 122 = 144. Check with n = 3: 152 = 225, and 9 + 72 + 144 = 225.
(x + 2)2 = x2 + 4.
✗ False. That leaves out the middle term. (x + 2)2 = x2 + 4x + 4. Check with x = 3: 52 = 25, but 32 + 4 = 13.
(x − 10)2 = x2 − 100.
✗ False. The expression x2 − 100 is the answer to (x + 10)(x − 10). Squaring the difference gives x2 − 20x + 100. Check with x = 3: (−7)2 = 49, but 9 − 100 = −91.
(4x − 1)(4x + 1) = 16x2 − 1.
✓ True. These are the same two terms, 4x and 1, with opposite signs, so the answer is (4x)2 − 12 = 16x2 − 1.
(a + b)(a − b) = a2 + b2.
✗ False. The middle terms cancel, but (−b)(b) = −b2, so the answer is a2 − b2. Check with a = 5 and b = 2: (7)(3) = 21 and 25 − 4 = 21, but 25 + 4 = 29.
(p − 4)2 = p2 − 8p + 16.
✓ True. The middle term is 2 × p × (−4) = −8p, and the last term is (−4)2 = +16. Check with p = 5: 12 = 1, and 25 − 40 + 16 = 1.
The last term of (x − 12)2 is −144.
✗ False. The last term is (−12)2 = +144. So (x − 12)2 = x2 − 24x + 144.
992 can be found as (100 − 1)2 = 10000 − 200 + 1 = 9801.
✓ True. With a = 100 and b = 1, a2 − 2ab + b2 = 10000 − 200 + 1 = 9801. Check: 99 × 99 = 9801.
(x + 9)(x − 4) = x2 − 36, because the middle terms cancel.
✗ False. The middle terms cancel only when the two factors use the same two terms. Here they are −4x and 9x, which combine to 5x. So (x + 9)(x − 4) = x2 + 5x − 36. Check with x = 3: 12 × (−1) = −12, but 9 − 36 = −27.
For real numbers a and b, (a + b)2 = a2 + b2 only when a = 0 or b = 0.
✓ True. The difference is (a + b)2 − (a2 + b2) = 2ab, which is 0 only when a = 0 or b = 0.
Because (x + 3)2 = x2 + 9x is true when x = 3, it is an identity.
✗ False. Both sides equal 36 at x = 3, but an identity must hold for every value. At x = 4 the left side is 49 and the right side is 52.
(−3x + 8)2 = 9x2 + 48x + 64, because the sign between the terms is a plus.
✗ False. The first term is −3x, so the double product is 2 × (−3x) × 8 = −48x. The answer is 9x2 − 48x + 64. Check with x = 3: (−9 + 8)2 = 1, and 81 − 144 + 64 = 1, but 81 + 144 + 64 = 289.
(x3 + 2)2 = x6 + 4x3 + 4.
✓ True. Square the first term: (x3)2 = x6. Double the product: 2 × x3 × 2 = 4x3. Square the last term: 4. Check with x = 3: (27 + 2)2 = 841, and 729 + 108 + 4 = 841.