Multiplying Binomials with Radicals
Multiply binomials containing radicals in Grade 9 algebra using the FOIL method or special product formulas, then simplify by combining like terms and reducing radicals where possible.
Key Concepts
Property Use the Distributive Property or FOIL method to multiply binomials containing radicals. For squaring a binomial: $$ (a b)^2 = a^2 2ab + b^2 $$ Explanation Remember your old pal FOIL? (First, Outer, Inner, Last). It’s the ultimate tool for multiplying two binomials, even with tricky radicals. This ensures every term gets multiplied by every other term. Watch out for the classic trap of just squaring the first and last parts of a binomial—that's a no go! Examples $ (5 + \sqrt{2})(3 \sqrt{2}) = 15 5\sqrt{2} + 3\sqrt{2} \sqrt{4} = 13 2\sqrt{2} $ $ (7 \sqrt{5})^2 = 7^2 2(7)\sqrt{5} + (\sqrt{5})^2 = 49 14\sqrt{5} + 5 = 54 14\sqrt{5} $.
Common Questions
How do you multiply binomials containing radicals?
Use FOIL just like with polynomial binomials. For (2 + √3)(4 - √3): F: 8, O: -2√3, I: 4√3, L: -(√3)(√3) = -3. Combine: 8 - 2√3 + 4√3 - 3 = 5 + 2√3.
How does the difference of squares apply to radical binomials?
For (a + √b)(a - √b), use the difference of squares: a² - (√b)² = a² - b. This eliminates the radical entirely. For (3 + √5)(3 - √5) = 9 - 5 = 4. Conjugate pairs like these always rationalize radicals.
What simplification steps follow multiplying radical binomials?
Multiply √a × √b = √(ab). Simplify any resulting radicals by factoring out perfect squares. Combine like radical terms (e.g., 3√2 + 5√2 = 8√2). Terms with different radicands (√2 and √3) cannot be combined.