explain binomial equation with proper visualisation. video should be slow and make it like explaining 6-7 grader with good animation, graphs and examples.
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Welcome to learning about binomial expressions! A binomial is a special type of algebraic expression that has exactly two terms. The word binomial comes from 'bi' meaning two, and 'nomial' meaning terms. Let's look at some examples to understand this better.
Notice how each binomial has exactly two parts separated by a plus or minus sign. The first term and second term can be numbers, variables, or combinations of both. Understanding binomials is the foundation for more advanced algebra topics we'll explore next!
Now let's learn how to add binomial expressions! When we add binomials, we need to combine like terms together. Like terms are terms that have the same variable and the same power. Let's work through our first example step by step.
Great! Now we combine the like terms. The x terms: 2x plus 4x equals 6x. The constant terms: 3 plus 1 equals 4. So our final answer is 6x plus 4. Let's try another example to make sure we understand the process.
Perfect! In this second example, we combined 3y and y to get 4y, and we combined negative 2 and positive 5 to get positive 3. Remember, when adding binomials, always look for like terms and add their coefficients together!
Now we're ready to learn about multiplying binomials! This is where algebra gets really exciting. When we multiply binomials, we use the distributive property. Let's start with a simple example to warm up.
Great! Now let's tackle multiplying two binomials. We can think of this like finding the area of a rectangle. Each binomial represents one dimension of the rectangle. Let me show you with x plus 2 times x plus 3.
Now we find the area of each section. Top left: x times x equals x squared. Top right: x times 3 equals 3x. Bottom left: 2 times x equals 2x. Bottom right: 2 times 3 equals 6. Finally, we add all these areas together to get our final answer!
Now let's learn the FOIL method! FOIL stands for First, Outer, Inner, Last. This gives us a systematic way to multiply any two binomials without missing any terms. Let's see how it works with our example.
Let's apply FOIL step by step. First terms: x times x equals x squared. Outer terms: x times 5 equals 5x. Inner terms: 3 times x equals 3x. Last terms: 3 times 5 equals 15. Now we combine like terms: 5x plus 3x equals 8x.
Let's try one more example to make sure we've got it! This time we have 2x plus 1 times x minus 4. Watch how we handle the negative sign carefully. The FOIL method works the same way, but we need to be careful with our signs when combining terms.
Welcome to the amazing world of binomial equations! A binomial is a special type of algebraic expression that has exactly two terms. Think of it like having two parts connected by a plus or minus sign. For example, x plus 5, or 2y minus 7. These are called binomials because 'bi' means two!
Now let's learn how to multiply binomials! We use a special method called FOIL. F-O-I-L stands for First, Outer, Inner, and Last. This tells us which terms to multiply together. Let's try multiplying x plus 3 times x plus 2. First, we multiply the first terms: x times x equals x squared.
Next, we multiply the outer terms: x times 2 equals 2x. Then the inner terms: 3 times x equals 3x. Finally, the last terms: 3 times 2 equals 6. Now we add all these terms together: x squared plus 2x plus 3x plus 6. We can combine like terms: 2x plus 3x equals 5x. So our final answer is x squared plus 5x plus 6!
Let's see the FOIL method visually! We draw arrows to connect the terms we need to multiply. The red arrow connects the first terms: x times x. The blue arrow connects the outer terms: x times 2. The green arrow connects the inner terms: 3 times x. And the purple arrow connects the last terms: 3 times 2.
This visual method helps us remember which terms to multiply together. The arrows make it clear and easy to follow. Each color represents one step of the FOIL process!
Time for practice! Let's try x plus 4 times x plus 1. Using FOIL: x times x gives us x squared, x times 1 gives us x, 4 times x gives us 4x, and 4 times 1 gives us 4. Adding them up: x squared plus x plus 4x plus 4. Combining like terms: x squared plus 5x plus 4.
Next example: y minus 2 times y plus 3. First terms: y times y equals y squared. Outer terms: y times 3 equals 3y. Inner terms: negative 2 times y equals negative 2y. Last terms: negative 2 times 3 equals negative 6. Our result: y squared plus 3y minus 2y minus 6, which simplifies to y squared plus y minus 6.
One more challenging example: 2x plus 1 times x minus 3. Using FOIL: 2x times x equals 2x squared, 2x times negative 3 equals negative 6x, 1 times x equals x, and 1 times negative 3 equals negative 3. This gives us 2x squared minus 6x plus x minus 3, which simplifies to 2x squared minus 5x minus 3. Great job practicing!
Now let's discover some amazing shortcuts! Certain binomial multiplications follow special patterns that can save us time. The first pattern is called the perfect square pattern. When we square a binomial like a plus b, we get a special result.
Let's see why this pattern works using our square diagram. We divide the square into four sections. We get a squared, plus ab, plus ab again, plus b squared. Notice we have two ab terms, so we get a squared plus 2ab plus b squared!
Let's practice with x plus 4 squared. Using our pattern: x squared plus 2 times x times 4 plus 4 squared equals x squared plus 8x plus 16. Now let's learn about the difference of squares pattern. When we multiply a plus b times a minus b, the middle terms cancel out!
These patterns are powerful shortcuts in algebra! The perfect square pattern helps us square binomials quickly, and the difference of squares pattern makes certain multiplications super easy. Remember these patterns - they'll save you lots of time!