Question

Find the quotient and the remainder. Check your work by verifying that Quotient \(\cdot\) Divisor \(+\) Remainder \(=\) Dividend Challenge Problem Without actually multiplying, determine the number of terms the simplified product will have $$ (x+y)(z+w)(x-y)(z-w) $$

Short answer

There will be 4 terms in the simplified product.

Step-by-step solution

- Recognize terms

Identify each term given in the expression \( (x+y)(z+w)(x-y)(z-w) \). There are four binomial factors in this product.

- Expand two binomials

First, expand two binomials \( (x+y)(x-y) \). This results in the difference of squares: \[ (x+y)(x-y) = x^2 - y^2 \].

- Expand the remaining two binomials

Next, expand the other two binomials \( (z+w)(z-w) \). This also results in the difference of squares: \[ (z+w)(z-w) = z^2 - w^2 \].

- Combine the results

Combine the results from Step 2 and Step 3 to get the final product: \[ (x^2 - y^2)(z^2 - w^2) \].

- Determine the number of terms

Observe that \( (x^2 - y^2)(z^2 - w^2) \) is a multiplication of two binomials, so the simplified product will have 4 terms. Therefore, there are 4 terms in the simplified product.

Key concepts

Binomial Expansion

Binomial expansion is a method used to multiply polynomials that contain two terms, often referred to as 'binomials.' When expanding binomials, each term in the first binomial is multiplied by each term in the second binomial. This works by using the distributive property. For instance, if you have ( x + y )( x - y ), you can distribute ( x ) over ( x - y ) and then ( y ) over ( x - y ).
This is essentially how we get x^2 from x * x and -y^2 from -y * y, giving us x^2 - y^2
Note how each term in ( x + y ) interacts with each term in ( x - y )..

Difference of Squares

The 'difference of squares' is a specific type of binomial product that has a unique pattern. It occurs when you multiply two binomials that have the same terms but different signs. For instance, ( x + y )( x - y ) results in ( x^2 - y^2 ).
This is a reliable pattern in algebra and is very useful when simplifying polynomial expressions.
The pattern works because the middle terms cancel out each other, leaving you with the subtraction of the squares of the individual terms.
We see this again in the simplified product: ( z + w )( z - w ) which simplifies to z^2 - w^2 .

Product of Binomials

Multiplying binomials involves distributing each term in the first binomial by each term in the second binomial. Let’s use ( x + y ) and ( z + w ) as an example.
We would do this:
x * z
x * w
y * z
y * w
This gives us the expanded polynomial: xz + xw + yz + yw
Here, each term in x + y interacts with each term in z + w .

Terms of a Polynomial

Terms of a polynomial are the distinct monomials that make up the polynomial.
In our problem, when you multiply (x^2 - y^2) with (z^2 - w^2), we need to distribute each term.
This means multiplying x^2 by both z^2 and - w^2.
So, breaking it down, we get four terms:
x^2 z^2, x^2 w^2, -y^2 z^2, and y^2 w^2.
These are the terms of the simplified polynomial product.
This added precision helps when verifying your final polynomial for correctness.