Binomials

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Binomial is a sum/difference of TWO monomials.

Both monomials are called terms.

Examples of binomials are:

  • $$${5}+{x}$$$
  • $$${2}{{x}}^{{2}}+{{y}}^{{3}}{x}$$$
  • $$${10}{x}{{y}}^{{3}}-{z}{{y}}^{{2}}{x}$$$

Examples of expressions, that are not binomials:

  • $$${x}+\frac{{1}}{{x}}$$$ (second term is not a monomial)
  • $$$\frac{{{2}-{y}}}{{x}}$$$ (division is not allowed)
  • $$${x}{{y}}^{{2}}+{z}-{2}$$$ (binomial can have only TWO terms)

Degree of the binomial is the largest number among degrees of its monomials.

For example, in binomial $$${4}{{x}}^{{2}}{{y}}^{{3}}-{9}{{z}}^{{8}}{{y}}^{{7}}$$$ first term has degree $$${2}+{3}={5}$$$ and second term has degree $$${8}+{7}={15}$$$. The largest of numbers 5 and 15 is 15. Thus, degree of the $$${4}{{x}}^{{2}}{{y}}^{{3}}-{9}{{z}}^{{8}}{{y}}^{{7}}$$$ is 15.

Exercise 1. Determine whether the following is a binomial: $$$\frac{{2}}{{5}}{x}-\frac{{3}}{{5}}{y}$$$?

Answer: yes.

Exercise 2. Determine whether the following is a binomial: $$$\frac{{1}}{{x}}+{2}{y}+{3}$$$?

Answer: no.

Exercise 3. Determine whether the following is a binomial: $$$\sqrt{{{3}}}{x}\cdot{y}+{5}{z}{y}$$$?

Answer: yes.

Exercise 4. Find degree of the following expression: $$$-{x}{y}+{5}$$$?

Answer: 2.

Exercise 5. Find degree of the following expression: $$$-{{p}}^{{3}}{{q}}^{{9}}+{5}{{z}}^{{3}}{{y}}^{{4}}{{p}}^{{{10}}}$$$?

Answer: 17.