Answer:
The distributive property is a helpful rule in algebra that allows us to multiply polynomials efficiently. It states that when we multiply a sum or difference of terms by a number or expression, we can distribute that number or expression to each term individually. This property is especially useful when multiplying polynomials, as it allows us to break down the problem into simpler steps.
Let's explore how the distributive property helps us multiply the given polynomials:
1. (a+b)^2:
To find the product of (a+b)^2, we can use the distributive property. This expression can be rewritten as (a+b) * (a+b). To simplify, we distribute the terms in the first polynomial, (a+b), to each term in the second polynomial, (a+b):
(a+b) * (a+b) = a * (a+b) + b * (a+b)
Now, we simplify further by applying the distributive property to each term:
a * (a+b) + b * (a+b) = a * a + a * b + b * a + b * b
Combining like terms, we get:
a^2 + ab + ab + b^2
Simplifying once more, we find the final product:
(a+b)^2 = a^2 + 2ab + b^2
2. (a-b)^2:
Similarly, to multiply (a-b)^2, we follow the same steps. Rewrite the expression as (a-b) * (a-b), then distribute each term:
(a-b) * (a-b) = a * (a-b) - b * (a-b)
Apply the distributive property:
a * (a-b) - b * (a-b) = a * a - a * b - b * a + b * b
Combine like terms:
a^2 - ab - ab + b^2
Simplify:
(a-b)^2 = a^2 - 2ab + b^2
3. (a-b) * (a+b):
To multiply (a-b) * (a+b), we can apply the distributive property directly:
(a-b) * (a+b) = a * (a+b) - b * (a+b)
Distribute each term:
a * (a+b) - b * (a+b) = a * a + a * b - b * a - b * b
Combine like terms:
a^2 + ab - ab - b^2
Simplify:
(a-b) * (a+b) = a^2 - b^2
In summary, the distributive property allows us to multiply polynomials by distributing each term of one polynomial to each term of the other. By applying this property, we can simplify the multiplication process and obtain the final products of (a+b)^2, (a-b)^2, and (a-b) * (a+b). The resulting products differ due to the different terms and their combinations in each expression.
Explanation:
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