Polynomials are expressions that consist of one or more terms. Terms are values consisting of constants, variables, and exponents. The different terms in a polynomial can be seperated by various mathematical operations such as addition and subtraction.
Different polynomials can be classified based on how many terms they have. An expression with only one term is called a monomial. An expression with two terms is considered a binomial. An expression with three terms is classified as a trinomial. Any expression with more than 3 terms is simply referred to as a polynomial.
You can simplify a polynomial by first expanding it. This can be done in a couple of different ways. The first way is using BEDMAS or PEDMAS (short for Brackets/Parentheses, Exponents, Division, Multiplication, Addition, Subtraction). This refers to the Order of Operations based on priority.
The second way is through using the FOIL (short for First, Outer, Inner, Last) that refers to the order you multiply a pair of binomials. You can further simplify the polynomial by collecting like terms. These are terms that contain the same variable and exponent.
Visit Expanding Polynomials to learn more about the different ways of polynomial expansion like FOIL!
Simplify the expression \(2a(-3a^2b^4)^3\), then identify the name of the polynomial.
First, we can expand the expression by raising the portion of the expression in the brackets (\(-3a^2b^4\)) to its exponent (\(3\)):
Next, we can simplify the expression by multiplying each of the terms in the brackets and collecting like terms:
\(= 2a(9a^{4}b^{8})(-3a^2b^4)\)
\(= 2a(-27a^{6}b^{12})\)
\(= -54a^7b^{12}\)
Therefore, we can determine that the expression simplified is \(\boldsymbol{-54a^7b^{12}}\).
We can also determine that this expression is a monomial since it only contains one term.
Visit Special Products to learn more about uncommon ways of expanding and simplifying polynomials!
First, we can expand the expression by multiplying the terms next to brackets:
\(= -2xy+5x^3+(2)(4)x^{1+1}y-xy+10x^{2+1})+(-6)(-3)x^2y\)
\(= \textcolor{red}{-2xy}+5x^3+\textcolor{blue}{8x^2y}-\textcolor{red}{xy}+10x^3+\textcolor{blue}{18x^2y}\)
Next, we can simplify the expression by adding and subtracting like terms. We can make this easier for ourselves by first placing all the like terms next to each other:
\(= \textcolor{red}{-2xy}-\textcolor{red}{xy}+\textcolor{blue}{8x^2y}+\textcolor{blue}{18x^2y}+5x^3+10x^3\)
\(= \textcolor{red}{-3xy}+\textcolor{blue}{26x^2y}+15x^3\)
Therefore, we can determine that the expression simplified is \(\boldsymbol{-3xy+26x^2y+15x^3}\).
We can also determine that this expression is a trinomial since it contains three terms.
First, we can substitute \(r\) and \(h\) into the expression with their respective values:
Next, we can simplify the expression by expanding it using FOIL:
\(V = \cfrac{π(2+x)(2+x)(2x-3)}{3}\)
\(V = \cfrac{π[(2)(2)+(2)(x)+(2)(x)+(x)(x)](2x-3)}{3}\)
\(V = \cfrac{π(4+2x+2x+x^2)(2x-3)}{3}\)
\(V = \cfrac{π(4+4x+x^2)(2x-3)}{3}\)
\(V = \cfrac{π((4)(2x)+(4)(-3)+(4x)(2x)+(4x)(-3)+(x^2)(2x)+(x^2)(-3))}{3}\)
\(V = \cfrac{π(\textcolor{blue}{8x}-12+\textcolor{green}{8x^2}-\textcolor{blue}{12x}+\textcolor{red}{2x^3}-\textcolor{green}{3x^2})}{3}\)
Finally, we can further simplify the expression by collecting like terms:
\(V = \cfrac{π(\textcolor{red}{2x^3}+\textcolor{green}{8x^2}-\textcolor{green}{3x^2}+\textcolor{blue}{8x}-\textcolor{blue}{12x}-12)}{3}\)
\(V = \cfrac{π(\textcolor{red}{2x^3}+\textcolor{green}{5x^2}-\textcolor{blue}{4x}-12)}{3}\)
Therefore, we can determine that the simplified expression for the Volume of the cone is \(\boldsymbol{\cfrac{π(\textcolor{red}{2x^3}+\textcolor{green}{5x^2}-\textcolor{blue}{4x}-12)}{3}}\).