📚 Topic Summary
Rationalizing binomial denominators involves eliminating radicals or imaginary numbers from the denominator of a fraction when the denominator contains two terms. This is achieved by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate is created by changing the sign between the two terms in the binomial. By multiplying by the conjugate, we leverage the difference of squares pattern, $(a+b)(a-b) = a^2 - b^2$, which eliminates the radical or imaginary part from the denominator.
For example, to rationalize the denominator of $\frac{1}{\sqrt{2} + 1}$, we multiply both the numerator and denominator by the conjugate $\sqrt{2} - 1$, resulting in $\frac{\sqrt{2} - 1}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1$.
🧠 Part A: Vocabulary
Match the term to its definition:
- Term: Rationalize
- Term: Binomial
- Term: Denominator
- Term: Conjugate
- Term: Radical
Definitions:
- The bottom part of a fraction.
- An expression that contains a root symbol.
- To eliminate radicals or imaginary numbers from the denominator.
- An expression with two terms.
- The binomial formed by changing the sign between two terms in a binomial.
| Term |
Definition |
| 1. Rationalize |
|
| 2. Binomial |
|
| 3. Denominator |
|
| 4. Conjugate |
|
| 5. Radical |
|
✏️ Part B: Fill in the Blanks
To rationalize a binomial denominator, we multiply both the numerator and the denominator by the ________ of the denominator. This eliminates the ________ or ________ numbers from the denominator by using the difference of ________ pattern. For example, to rationalize $\frac{1}{2 + \sqrt{3}}$, we multiply by ________.
🤔 Part C: Critical Thinking
Explain why multiplying by the conjugate eliminates the radical in the denominator when the denominator is in the form $a + \sqrt{b}$.