Simplify (a^2+b^2)(c^2+d^2)
The problem is asking for the simplification of the expression (a^2+b^2)(c^2+d^2). This entails performing the multiplication of two binomials in such a way that the result is expressed as a single polynomial, by expanding the product using the distributive property (also known as the FOIL method for binomials), which involves multiplying each term in the first binomial by each term in the second binomial and then combining like terms, if any.
$\left(\right. a^{2} + b^{2} \left.\right) \left(\right. c^{2} + d^{2} \left.\right)$
Answer
Solution:
Step 1:
Utilize the distributive law to expand the expression: $a^2(c^2 + d^2) + b^2(c^2 + d^2)$.
Step 2:
Further apply the distributive law to the first term: $a^2c^2 + a^2d^2 + b^2(c^2 + d^2)$.
Step 3:
Complete the expansion by distributing the second term: $a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2$.
Knowledge Notes:
The problem involves simplifying a product of two binomials where each binomial is a sum of squares. To solve this, we use the distributive property (also known as the distributive law of multiplication over addition), which states that for all real numbers $a$, $b$, and $c$, the equation $a(b + c) = ab + ac$ holds true. The distributive property allows us to multiply each term in the first binomial by each term in the second binomial, resulting in four terms.
In this case, the distributive property is applied in a step-by-step manner:
First, we distribute $a^2$ over the second binomial $(c^2 + d^2)$, which results in two terms: $a^2c^2$ and $a^2d^2$.
Then, we distribute $b^2$ over the same binomial $(c^2 + d^2)$, which also results in two terms: $b^2c^2$ and $b^2d^2$.
Finally, we combine all four terms to obtain the simplified expression.
This process does not result in further simplification since there are no like terms to combine. The final expression is the sum of four distinct terms, each representing the product of a term from the first binomial and a term from the second binomial.
