Introduction to Polynomials – Naming Polynomials by Number of Terms

Introduction to Polynomials – Naming Polynomials by Number of Terms exercise

• Identify which of the following statements about monomials are true.

  Hints

  A power is short-hand notation for a product:

  $x^n=\underbrace{x\times ...\times x}_{\text{n times}}$.

  We can see that $x^1$ has only one factor, $x$.

  You can divide two powers with the same base by subtracting their exponents:

  ${\large\frac{x^{\large 4}}{x^{\large 2}}}=x^{(4-2)}=x^2$.

  Solution

  Monomials are coefficients times variables raised to non-negative, integer exponents.

  $~$

  • The variable $x$ itself is a monomial. This is because $x=1x^1$. When a variable is raised to the power of $1$, we don't need to write the exponent.

  • Any constant is a monomial. This is because $x$ raised to the power of $0$ is equal to $1$, so $5x^0=5$.

• Find the term representing the cost of Bella's quinceanera.

  Hints

  Keep the definition of a monomial in mind:

  Monomials are coefficients times variables raised to non-negative, integer exponents.

  A term consisting of two monomials is called a binomial, and a term consisting of three monomials is called a trinomial.

  Constants or variables standing alone are also minomials; for instance, we have:

  • $5=5x^0$
  • $x=1x^1$

  Solution

  Bella wakes up super early to plan her quinceanera! A very exciting time in a young woman's life.

  She starts to put together a list of party expenses:

  • The party venue will cost 300 dollars, no question.
  • Invitations will cost 4 dollars a pop. Letting $x$ be the number of invitations, the total cost for invitations will be $4x$ all in all.
  • The DJ wants 50 dollars per hour. Letting $y$ be the number of hours the DJ will play, she knows that the DJ will cost $50y$ in total.
  • Last but not least: Her dress. She absolutely wants to look like a movie star... and there is no telling what the price of looking like fame will be, she just lets $z$ represent the price of her look.

  Each of the terms above are monomials.

  Last she adds all the costs together to get the polynomial, $300+4x+50y+z$.

• Analyze the following terms.

  Hints

  Monomials are coefficients times a variable or variables that are raised to non-negative, integer exponents.

  Binomials are terms with two monomials.

  Pay attention: variables raised to negative or rational powers are not monomials.

  Solution

  Monomials are coefficients times a variable or variables that are raised to non-negative, integer exponents.

  The following terms are monomials:

  • $34x$
  • $-4y^2$
  • $123$

  $~$

  Neither $4y^{-2}$ nor $123x^{0.5}$ are monomials, as the exponents aren't positive integers.

  $~$

  The following terms are binomials, as they are made up of two monomials:

  • $34x-4y^2$
  • $34x+123$
  • $-4y^2+123$

  $~$

  Since $4y^{-2}$ is not a monomial, we know that $34x+4y^{-2}$ cannot be a binomial.

  $~$

  $34x-4y^2+123$ is a trinomial, so it is neither a binomial nor a monomial. Adding or subtracting some further monomials leads to polynomials. For instance, $4x^3+2x^2-x-6$ is a polynomial as it is a term made up of more than three monomials... so it is also neither a binomial nor a monomial.

• Establish the polynomial representing the total cost of Bella's next shopping trip.

  Hints

  Constants or variables are monomials.

  • A term consisting of two monomials is a binomial.

  • A term consisting of three monomials is a trinomial.

  • A term consisting of more than three monomials is a polynomial.

  Solution

  Before Bella embarks on a day shopping in town, she needs to figure out how much money she will need for the trip!

  • She needs $10$ dollars for a bus ticket.
  • She plans to buy a new pair of jeans, but doesn't know how much they would cost. So she lets the price be $x$, a variable.
  • She then always goes to her favortie ice cream parlor, which sells ice cream at 1.5 dollars a scoop. She's still not sure exactly how many scoops she will want... so she decides to represent the number of scoops she buys by $y$. This gives her a monomial, $1.5y$, representing the cost of the ice cream.
  • She also needs three new school notebooks, which cost 2 dollars each... or in other words, $3\times 2=6$ dollars.

  She adds all of the costs to together to get the polynomial $10+x+1.5y+6$.

• Decide if the given term is a monomial, binomial, trinomial, or polynomial.

  Hints

  Monomials are coefficients times variables raised to non-negative, integer exponents.

  $-\frac13x^2$ is an example of a monomial.

  • Mono stands for one.
  • Bi stands for two.
  • Tri stands for three.
  • Poly stands for many.

  Solution

  Monomials are coefficients times variables raised to non-negative, integer exponents.

  • A term of two monomials is a binomial.
  • A term of three monomials is a trinomial.
  • A term of more than three monomials is a polynomial.

  Let's look at some examples:

  Monomials

  • $4x^3$
  • $-\frac13x^2$
  • $x$
  • $5$
  • $1$

  Binomials

  • $4x^3-\frac13x^2$
  • $x+5$

  Trinomials

  • $4x^3-\frac13x^2+x$
  • $4x^3+x+1$

  Polynomials

  • $4x^3-\frac13x^2+x+5$
  • $4x^3-\frac13x^2+x+1$

• Simplify the monomials and then add them together to make a polynomial.

  Hints

  Think about the meaning of $x^n$:

  $x^n=\underbrace{x\times ...\times x}_{\text{n times}}$.

  For example, we have that $5x^1=5x$.

  Solution

  Let's start by simplifying the monomials. We have:

  • $1x^5=x^5$
  • $22x^0=22\times 1=22$
  • $3x^1=3x$
  • $1x^3=x^3$

  $~$

  Next we add all of the monomials together to get:

  $x^5+22+3x+x^3$.

  While writing a sum of monomials in any order you choose is correct, conventionally, such summands are arranged from highest to lowest degree:

  $x^5+x^3+3x+22$.