Building Sentences: Polynomials

A polynomial is an equation with a mix of numbers and variables. The terms of a polynomial are separate by operations. If a polynomial only has one term it is called a monomial. If it has two terms it is called a binomial. If it has three terms it is called a trinomial. All other polynomials are simply called a polynomial.

A polynomial has two parts. The real number part is called the coefficient and the letter parts are called the variable. In the polynomial 3x+2, x is a variable and 3 and 2 are the coefficients. Some of the parts of 3x+2 are not shown. The exponent on the x is 1. There is a variable that goes with the coefficient 2; remember from our discussion of exponents that anything raised to the 0 power is 1, so 2 is actually 2×1=2×xº. The terms where all of the variables are 0 powered are called constants.

The degree of a polynomial is the highest exponent of the polynomial. If x³ is the highest exponent in the polynomial we say it is a third degree polynomial. A polynomial of degree 1 is called a linear equation. A polynomial of degree 2 is called a quadratic equation. Degree 3 polynomials are called cubics. A 4th degree polynomial is called a quartic and a 5th degree polynomial is called a quintic. A linear equation draws a line when we graph it. All even degree polynomials graph a parabola. All odd degree polynomials approximate a line. I can't show you pictures in this blog, so we won't be able to cover graphing of polynomials. But a parabola looks like a u. All even degree polynomials enter a graph going one direction, then change directions at some point and leave the graph going in the opposite direction. All odd degree polynomials enter the graph in one direction and leave the graph in the same direction. Polynomials of degree higher than 2 sometimes bounce in the middle.

It is customary to write polynomials in descending degree order with coefficients first followed by variables in alphabetical order followed by radicals.

When adding two polynomials, you add only the terms that have the same variables with the same exponents. You simply add the coefficients.
(x²y³+xy)+(2x²y³+xy)=
((1)x²y³+(1)xy)+(2x²y³+(1)xy)=
(1+2)x²y³+(1+1)xy=
3x²y³+2xy.

If you're subtracting a polynomial from another polynomial, it's best to distribute the subtraction to all of the terms of the second polynomial at the start. You change the subtraction operation to an addition operation and change the sign of each of the terms of the second polynomial.
(2x²+3x+1)-(x²+x-1)=
(2x²+3x+1)+(-x²-x+1)
Now you drop the parenthesis and combine like terms.
2x²-x²+3x-x+1+1=
(2-1)x²+(3-1)x+(1+1)=
x²+2x+2

When multiplying two polynomials, we have to use the distributive property of real numbers. Every term in the first polynomial has to be distributed to each term of the second polynomial. Let's first multiply a polynomial by a constant.
3(x⁴+2x³+x²+3x+4)=
(3×x⁴)+(3×2x³)+(3×x²)+(3×3x)+(3×4)=
3x⁴+(3×2)x³+3x²+(3×3)x+12=
3x⁴+6x³+3x²+9x+12.

Next, let's multiply two binomials. We use distribution twice.
(x²y+2)×(xy²-1)=
x²y×(xy²-1)+2×(xy²-1)
Notice how the mutliplication sign and the second binomial get inserted into the first binomial. Now we distribute again.
(x²y×xy²)+(x²y×-1)+(2×xy²)+(2×-1)
Watch out for negative signs. Make sure they distribute with the number. Now, we multiply everything out. Remember to add exponents when the letters are the same.
x²⁺¹y¹⁺²-x²y+2xy²-2=
x³y³-x²y+2xy²-2

We'll leave dividing two polynomials until later.