OP - Addition, Subtraction, and Multiplication of Polynomial Expressions and Functions Lesson

Addition, Subtraction, and Multiplication of Polynomial Expressions and Functions

Adding and Subtracting Polynomials

Polynomials can have operations performed on them just like numbers. We'll start with adding and subtracting.

When adding polynomials, you will add like terms together. Remember that like terms have the exact same variable and exponent.

If you know how to add polynomials, you will be able to subtract them! In adding polynomials you can add one of two ways...horizontally or vertically. The same is true for subtraction. Also, subtraction of polynomials can be illustrated as adding the opposite. Just like in adding polynomials, you must subtract like terms.

Watch these videos to review the addition and subtraction of polynomials.

Now it is time for you to complete the Adding & Subtracting Polynomials Self-Assessment.

Multiplying Polynomials

To multiply polynomials, the distributive property is used; which is...for all real numbers a, b, and c, a(b + c) = ab + ac and (b + c)a = ba + ca. This also is true for subtraction. (Remember, when multiplying like bases, you add the exponents together: $a^m+a^n=a^{m+n}$). When you are multiplying 2 binomials, you distribute using a distributive method called "FOIL".

Watch these videos to review the multiplication of polynomials.

Binomial Theorem

There are times when multiplying a binomial by a binomial is very expedient. This process is referred to as "FOIL-ing" or "distributing twice." What if we wanted to multiply a binomial by itself more than twice? Multiplying $\left(w-3\right)^3=\left(w-3\right)\left(w-3\right)\left(w-3\right)$ could take a long time, but it can be done "fairly quickly". Finding $\left(w-3\right)^{11}$ would be very tedious and leave a lot of room for error.

Binomial Coefficients in Pascal's Triangle

Pascal's triangle can be used to determine the coefficients in binomial expansions.

For example:

$\left(x+y\right)^0=1$

$\left(x+y\right)^1=x+y=1x^1y^0+1x^0y^1$

$\left(x+y\right)^2=x^2+2xy+y^2=1x^2y+2x^1y^1+1x^0y^2$

$\left(x+y\right)^3=x^3+3x^2y+3xy^2+y^3=1x^3y^0+3x^2y^1+3x^1y^2+1x^0y^3$

Notice that the coefficients of $\left(x+y\right)^n$ correspond to numbers in row n of Pascal's Triangle.

When raising a binomial to a power, the degree of each term will be the same as the power of the binomial. For example, in the problem 

$\left(x+y\right)^6=x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6$ if you add the powers in each.

 

Now it is time for you to complete the Multiplying Polynomials Self-Assessment.

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