Question

I NEED HELP ASAP!!!!

Explain how the distributive property helps us multiply the following polynomials and why and how the final products differ:
(a + b)^2, (a - b)^2, and (a - b)(a + b).
(I need an explanation, not just the answer, please.)

Answer 1

Explanation:

The distributive property states that for any three real numbers a, b, and c, we have:
a(b + c) = ab + ac

This property is very useful when multiplying polynomials, because it allows us to distribute each term in one polynomial to every term in the other polynomial.

For example, to square the binomial (a + b), we can use the distributive property twice:
(a + b)^2 = (a + b)(a + b)
= a(a + b) + b(a + b)
= a^2 + ab + ab + b^2
= a^2 + 2ab + b^2

We can see that the distributive property allows us to multiply each term in the first polynomial by each term in the second polynomial. We then group like terms and simplify to get the final product.

Similarly, to square the binomial (a - b), we can use the same process:
(a - b)^2 = (a - b)(a - b)
= a(a - b) - b(a - b)
= a^2 - ab - ab + b^2
= a^2 - 2ab + b^2

Again, we see that the distributive property enables us to distribute the terms in one polynomial to the other polynomial. We then group like terms and simplify to get the final product.
To find the product of (a - b)(a + b), we can use the distributive property again:
(a - b)(a + b) = a(a + b) - b(a + b)
= a^2 + ab - ab - b^2
= a^2 - b^2

This time, we see that the product is a difference of squares, which means that it can be factored further. In general, the product of two binomials that are conjugates of each other (i.e. have the same terms, but with one sign changed) will be a difference of squares.