When dealing with algebraic expressions, simplifying them is a crucial step in solving equations and understanding the underlying mathematical concepts. One such expression that may seem daunting at first is y(1+2x)^3. However, with the right approach, it can be simplified into a more manageable form. In this article, we will break down the process of simplifying y(1+2x)^3 into 5 easy steps, making it accessible to anyone with a basic understanding of algebra.
Key Points
- Understanding the binomial expansion formula is essential for simplifying expressions like y(1+2x)^3.
- The binomial expansion formula is (a + b)^n = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k, where n is a positive integer and the sum is taken from k=0 to n.
- Applying the binomial expansion formula to (1+2x)^3 involves calculating the coefficients and powers of 1 and 2x for each term.
- Simplifying the expression requires combining like terms and applying basic algebraic rules.
- The simplified form of y(1+2x)^3 can be used in various algebraic and calculus applications.
Step 1: Understand the Binomial Expansion Formula
The binomial expansion formula is a fundamental concept in algebra that allows us to expand expressions of the form (a + b)^n. The formula is given by (a + b)^n = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k, where n is a positive integer and the sum is taken from k=0 to n. This formula will be our tool for simplifying y(1+2x)^3.
Applying the Formula to (1+2x)^3
To apply the binomial expansion formula to (1+2x)^3, we need to identify a, b, and n. In this case, a = 1, b = 2x, and n = 3. Plugging these values into the formula gives us (1+2x)^3 = ∑[3! / (k!(3-k)!)] * 1^(3-k) * (2x)^k, where the sum is taken from k=0 to 3.
| k | Term |
|---|---|
| 0 | 1^3 * (2x)^0 = 1 |
| 1 | 3 * 1^2 * (2x)^1 = 6x |
| 2 | 3 * 1^1 * (2x)^2 = 12x^2 |
| 3 | 1^0 * (2x)^3 = 8x^3 |
Step 2: Calculate Each Term
Using the formula, we calculate each term for k = 0, 1, 2, and 3. For k = 0, the term is 1^3 * (2x)^0 = 1. For k = 1, the term is 3 * 1^2 * (2x)^1 = 6x. For k = 2, the term is 3 * 1^1 * (2x)^2 = 12x^2. For k = 3, the term is 1^0 * (2x)^3 = 8x^3.
Combining the Terms
Now that we have calculated each term, we can combine them to get the expanded form of (1+2x)^3. This gives us (1+2x)^3 = 1 + 6x + 12x^2 + 8x^3.
Step 3: Simplify y(1+2x)^3
To simplify y(1+2x)^3, we multiply y by each term in the expanded form of (1+2x)^3. This gives us y(1+2x)^3 = y(1 + 6x + 12x^2 + 8x^3) = y + 6yx + 12yx^2 + 8yx^3.
Distributive Property
The distributive property of multiplication over addition allows us to multiply y by each term in the expanded form of (1+2x)^3. This property is essential for simplifying expressions like y(1+2x)^3.
Step 4: Combine Like Terms
In this case, there are no like terms to combine, as each term has a unique power of x. Therefore, the simplified form of y(1+2x)^3 is y + 6yx + 12yx^2 + 8yx^3.
No Further Simplification
Since there are no like terms to combine, the expression y + 6yx + 12yx^2 + 8yx^3 is the simplest form of y(1+2x)^3. This expression can be used in various algebraic and calculus applications.
Step 5: Verify the Simplification
To verify the simplification, we can substitute specific values of x and y into the original expression y(1+2x)^3 and the simplified expression y + 6yx + 12yx^2 + 8yx^3. If the two expressions are equal for all values of x and y, then the simplification is correct.
What is the binomial expansion formula?
+The binomial expansion formula is (a + b)^n = ∑[n! / (k!(n-k)!)] * a^(n-k) * b^k, where n is a positive integer and the sum is taken from k=0 to n.
How do I apply the binomial expansion formula to (1+2x)^3?
+To apply the binomial expansion formula to (1+2x)^3, identify a = 1, b = 2x, and n = 3, and plug these values into the formula.
What is the simplified form of y(1+2x)^3?
+The simplified form of y(1+2x)^3 is y + 6yx + 12yx^2 + 8yx^3.
In conclusion, simplifying y(1+2x)^3 involves applying the binomial expansion formula, calculating each term, combining like terms, and verifying the simplification. By following these steps, we can simplify y(1+2x)^3 into the form y + 6yx + 12yx^2 + 8yx^3, which can be used in various algebraic and calculus applications.