When multiplying two polynomials, what fundamental property do you use repeatedly?
The fact that multiplication distributes over addition, otherwise known as the "distributive property
That is correct.
When multiplying two polynomials, the fundamental property that you use repeatedly is the distributive property. This property states that for any given real numbers or variables a, b, and c:
a * (b + c) = ab + ac
When applied to polynomials, the distributive property allows you to multiply each term of one polynomial with every term of the other polynomial. This process is repeated for each term in both polynomials, leading to the overall multiplication of the two polynomials. Let's see an example to illustrate this.
Suppose we have two polynomials:
(a + b) * (c + d)
To multiply these two polynomials, we need to apply the distributive property repeatedly. We start by multiplying the first term of the first polynomial (a) with every term of the second polynomial (c and d):
a * c + a * d
Next, we multiply the second term of the first polynomial (b) with every term of the second polynomial (c and d):
b * c + b * d
Finally, we combine all the resulting terms:
(ac + ad) + (bc + bd)
This is the simplified form of the product of the two polynomials. By applying the distributive property repeatedly, we multiplied each term in one polynomial with every term in the other polynomial to obtain the final result.