Question

Explain what a polynomials is and identify the different parts of a polynomial.
Explain the different labels used to categorize polynomials
Explain how addition and subtraction of polynomials is accomplished
When multiplying polynomials, we are taking every term of one polynomial and multiplying them by every term of the second polynomial, then collecting like terms explain how foil helps us to accomplish this and what category of polynomials foil applies to.
There are tow special case products where the product takes on special patterns explain these two cases and when they occur.
For each of the two special cases, illustrate your explanation with an appropriate exercise from the ( a+b)2=a2+2ab+b2 )(a-b)2=a2-2ab+b2 ) showing how the multiplication is performed using the special pattern formula then also showing the same result using foil.

Answer 1

A polynomial is an expression of more than one term. An expression is considered a polynomial when is has more than one term, otherwise, it would be called a monomial. These can be combined together through multiplication, addition and subtraction only. (Meaning no division or fractions)

Ex.

$3 x^{2} + 3x - 3$

 x is a variable (There can be more than 1 variable in a term. Ex. 3xy, 4xyz, 4ab)

*A variable may be represented by letters.

2 is an exponent

3 is a constant

Those are the parts of a polynomial.

Polynomials can be categorized depending on the number of terms and their degree.

A polynomial with two terms is called a binomial. If it has three terms it is called a trinomial. If the expression has more than 3 terms, they are generally called polynomials.

A polynomial can be categorized by degree as well. You can determine the degree of a polynomial by looking at the term that has the highest exponent.

Using the example above, you can categorize the polynomial as a 2nd degree trinomial because 2 is the highest exponent and it has three terms.

When you add and subtract polynomials you need to take note of the variables. You can only subtract and add like terms, which means that the variables and the exponents are the same.

Ex.

$( 2x^{3} + 2 y^{2} + x + 1) + (4 x^{2} - y^{2} + y + 2)$

When you add these two polynomials, you can disregard the parentheses because according to the associative property of addition, no matter how you group the terms, the answer will be the same.

Like mentioned before you can only add and subtract like terms. It would be easier if you just group like terms together by rearranging the expression. Do not forget that the sign or operation comes along with them.

$2 x^{3} + 2 y^{2} - y^{2} + 4 x^{2} + x + y + 2 + 1$

Now combine the like terms.

$2 x^{3} + y^{2} + 4 x^{2} + x + y + 3$

Notice that we retained the terms $2 x^{3}$ , $4 x^{2}$, x and y, this is because they have no similar terms.

FOIL method is used when multiplying 2 BINOMIALS. Remember that a binomial is an expression with 2 terms.

FOIL means:

FIRST term: first terms of each binomial.

OUTSIDE term: The two outer terms when taking the equation as a whole.

INSIDE term: The two inner terms when taking the equation as a whole.

LAST term: Last term of each binomial (2nd term of each binomial)

To get the answer, you need to multiply them with their corresponding term.

Ex. (2x+3)(x-4)

F:  2x and x            (2x)(x) = $2x^{2}$     

O: 2x and -4    (2x)(-4) = -8x

I: 3 and x          (3)(x)   = 3x

L: 3 and -4           (3)(-4)   = -12

Resulting expression:

$2 x^{2} - 8x + 3x -12$         -8x and 3x are similar or like terms, so you can combine them

$2 x^{2} - 5x -12$

When doing multiplication with binomials, there are two special cases you can consider doing, which follow a pattern. The first is multiplying sum and difference.

The condition where you can apply the first special case is the first term needs to be the same and the second term are additive inverses.

(a+b)(a-b)

The resulting expression follows this pattern $a^{2} - b^{2}$

Ex. (x+3)(x-3) = $x^{2} - 3^{2}$ or $x^{2} -9$

You can use FOIL to check your answer:

F: (x)(x) = $x^{2}$

O: (x)(-3) = $-3x$

I: (3)(x) = $3x$

L: (3)(-3) = $-9$

Arrange the expression:

$x^{2} - 3x + 3x - 9$      Combining -3x+3x = 0 

$x^{2} -9$

The next special case is squaring a binomial and there are two scenarios that you can consider. 

$(a+b)^{2}$ and $(a-b)^{2}$

The resulting expression follows a certain pattern for each:

$(a+b)^{2}$ = $a^{2} + 2ab + b^{2}$

$(a-b)^{2}$ = $a^{2} - 2ab + b^{2}$

Let's use an example of each to demonstrate this and check with FOIL:

$(a+b)^{2}$

$(2x+4)^{2}$

a = 2x     b = +4

Let's insert that into our pattern:

$a^{2} + 2ab + b^{2}$

 $2x^{2} + 2(2x)(4) + 4^{2}$

Simplify the expression:

$2x^{2} + 16x + 4^{2}$

$4x^{2} + 16x + 16$

Let's check with FOIL

$(2x+4)^{2}$ = $(2x+4)(2x+4)$

F: (2x)(2x) = $4x^{2}$

O: (2x)(4) = $8x$

I: (4)(x) = $8x$

L: (4)(4) = $16$

Let's arrange the terms:

$4x^{2} + 8x + 8x + 16$      Combine the like terms
$4x^{2} + 16x + 16$   It's the same.

Now let's use the second scenario:

$(a-b)^{2}$
$(2x-4)^{2}$

a = 2x     b = -4

Let's insert that into our pattern:

$a^{2} - 2ab + b^{2}$

 $2x^{2} - 2(2x)(-4) + (-4)^{2}$

Simplify the expression:

$2x^{2} - 16x + (-4)^{2}$

$4x^{2} - 16x + 16$

Let's check with FOIL

$(2x+4)^{2}$ = $(2x-4)(2x-4)$

F: (2x)(2x) = $4x^{2}$

O: (2x)(-4) = $-8x$

I: (-4)(x) = $-8x$

L: (-4)(-4) = $16$

Let's arrange the terms:

$4x^{2} - 8x - 8x + 16$      Combine the like terms
$4x^{2} - 16x + 16$   It's the same.