Discover the Magic of Polynomial Square Roots ✨
Let's learn how to find square roots of polynomials step by step!
Make sure the polynomial is arranged in descending order of exponents. If any term is missing, use a 0 coefficient as a placeholder.
Starting from the right, pair the terms in groups of two. For odd-degree polynomials, the leftmost term will be alone.
Similar to numerical square roots, find terms that when squared and subtracted, bring down the next pair.
Double the current result, find the next digit, multiply, subtract, and bring down until all terms are exhausted.
Multiply your result by itself to verify it matches the original polynomial.
1. Polynomial is already in descending order: x⁴ − 12x³ + 42x² − 36x + 9
2. Find square root:
3. Verification: (x² - 6x + 3)² = x⁴ - 12x³ + 42x² - 36x + 9 ✓
∴ √(x⁴ − 12x³ + 42x² − 36x + 9) = x² - 6x + 3
1. Rearrange in descending order: 4x⁴ - 28x³ + 37x² + 42x + 9
2. Find square root:
3. Verification: (2x² - 7x - 3)² = 4x⁴ - 28x³ + 37x² + 42x + 9 ✓
∴ √(4x⁴ - 28x³ + 37x² + 42x + 9) = 2x² - 7x - 3
1. Polynomial is in descending order: 16x⁴ + 0x³ + 8x² + 0x + 1
2. Find square root:
3. Verification: (4x² + 1)² = 16x⁴ + 8x² + 1 ✓
∴ √(16x⁴ + 8x² + 1) = 4x² + 1
1. Polynomial is already in descending order: 121x⁴ − 198x³ − 183x² + 216x + 144
2. Find square root:
3. Verification: (11x² - 9x - 12)² = 121x⁴ - 198x³ - 183x² + 216x + 144 ✓
∴ √(121x⁴ − 198x³ − 183x² + 216x + 144) = 11x² - 9x - 12
1. Assume the square root is of form (2x² + px + q)
2. (2x² + px + q)² = 4x⁴ + 4px³ + (p²+4q)x² + 2pqx + q²
3. Compare coefficients:
- 4p = -12 ⇒ p = -3
- p² + 4q = 37 ⇒ 9 + 4q = 37 ⇒ q = 7
- 2pq = b ⇒ 2(-3)(7) = b ⇒ b = -42
- q² = a ⇒ 49 = a
∴ a = 49, b = -42
1. Assume the square root is of form (mx² + px + 10) since √100 = 10
2. (mx² + px + 10)² = m²x⁴ + 2mpx³ + (p²+20m)x² + 20px + 100
3. Compare coefficients:
- 20p = 220 ⇒ p = 11
- p² + 20m = 361 ⇒ 121 + 20m = 361 ⇒ m = 12
- 2mp = b ⇒ 2×12×11 = b ⇒ b = 264
- m² = a ⇒ 144 = a
∴ a = 144, b = 264
1. Assume the square root is of form (6x² + px + q)
2. (6x² + px + q)² = 36x⁴ + 12px³ + (p²+12q)x² + 2pqx + q²
3. Compare coefficients:
- 12p = -60 ⇒ p = -5
- p² + 12q = 61 ⇒ 25 + 12q = 61 ⇒ q = 3
- 2pq = -m ⇒ 2(-5)(3) = -m ⇒ m = 30
- q² = n ⇒ 9 = n
∴ m = 30, n = 9
1. Assume the square root is of form (x² + px + 4) since √16 = 4
2. (x² + px + 4)² = x⁴ + 2px³ + (p²+8)x² + 8px + 16
3. Compare coefficients:
- 2p = -8 ⇒ p = -4
- p² + 8 = m ⇒ 16 + 8 = m ⇒ m = 24
- 8p = n ⇒ 8×(-4) = n ⇒ n = -32
∴ m = 24, n = -32
Try solving these problems yourself before revealing the solutions!