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**Explain the following concepts: a term, a coefficient, a constant term, and a polynomial.**

**A term:**

When numbers are added or subtracted, they are called terms. This – 4*x*² + 7*x* − 8

- is a sum of three terms: 4x
^{2}, 7x and -8.

**A variable: **

(we need also to explain the concept of “a variable” in order to explain correctly further the concept of “a polynomial”)

A variable is a symbol that takes on values. A value is a number. The expression considered above has a variable *x*. Thus if *x* has the value 1, then 4*x*² + 7*x* − 8 has the value 3. If *x* has the value 0, then 4*x*² + 7*x* − 8 has the value -8. A variable may be denoted by another letter than *x*; for instance *y* or *z*, or any other.

**A constant term:**

The constant term is the term in which the variable does not appear. It is called a “constant” term because, no matter what value one may put in for the variable *x*, that constant term will never change.

The expression 4x² + 7x − 8 has a constant term -8.

**A polynomial: **

A polynomial in a variable* x* is a sum of terms of the form *ax** ^{n}*, where

**What is the difference between a monomial, a binomial, and a trinomial?**

Certain polynomials have special names depending on the number of terms. A **monomial** is a polynomial that has one term, a **binomial** is a polynomial that has two terms, and a **trinomial** is a polynomial that has three terms.

**Explain what the degree of a term and the degree of a polynomial mean.**

The degree of a term is the exponent of the variable in that term. For example: the degree of 4*x*² is 2, the degree of 7*x* is 1. The degree of constant term is 0.

The degree of polynomial in one variable is the highest power of the variable in the polynomial. Hence, the degree of trinomial 4*x*² + 7*x* − 8 is 2.

** ****Give an example to show how to: **

**a. Combine like terms.
b. Add and subtract polynomials.**

** **

- When we say “combine like terms” – we sort and collect terms which are alike, then write a simplified expression.
**Example 1:**

3x – 6y + 5x – 5y = 3x + 5x – 6y – 5y3x and 5x are like terms;

6y and 5y are like terms.3x + 5x – 6y – 5y = (3x + 5x) – (6y + 5y) = 8x – 11y.

**Example 2:**

4(x² – 3x) – 2(x – 2) – (3 – x – x²)

Here we must distribute first

4(x² – 3x) – 2(x – 2) – (3 – x – x²) = 4x^{2} – 12x – 2x + 4 – 3 + x + x^{2} =

= 4x^{2} + x^{2} – 12x – 2x + x + 4 – 3

4x^{2} and x^{2} are like terms;

-12x, -2x, and x are like terms;

4 and -3 are like terms.

4x^{2} + x^{2} – 12x – 2x + x + 4 – 3 = (4x^{2} + x^{2}) + (-12x – 2x + x) + (4 – 3) = 5x^{2} – 13x + 1.

- First polynomial:x
^{4}+ 7x^{3}+ kx^{2}– 3.5x + 1

Second polynomial:** **ax^{3} – x^{2} + 2x – 0.5

**Adding: **

(x^{4} + 7x^{3} + kx^{2} – 3.5x + 1) + (ax^{3} – x^{2} + 2x – 0.5) =

= x^{4} + 7x^{3} + kx^{2} – 3.5x + 1 + ax^{3} – x^{2} + 2x – 0.5 =

= x^{4} + 7x^{3} + ax^{3} + kx^{2} – x^{2} – 3.5x + 2x + 1 – 0.5 =

= x^{4} + (7x^{3} + ax^{3}) + (kx^{2} – x^{2}) + (-3.5x + 2x) + (1 – 0.5) =

= x^{4} + (7 + a)x^{3} + (k – 1) x^{2} + (-1.5x) + 0.5 =

= x^{4} + (7 + a)x^{3} + (k – 1) x^{2} – 1.5x + 0.5

**Subtracting:**

(x^{4} + 7x^{3} + kx^{2} – 3.5x + 1) – (ax^{3} – x^{2} + 2x – 0.5) =

= x^{4} + 7x^{3} + kx^{2} – 3.5x + 1 – ax^{3} + x^{2} – 2x + 0.5 =

= x^{4} + 7x^{3} – ax^{3} + kx^{2} + x^{2} – 3.5x – 2x + 1 + 0.5 =

= x^{4} + (7x^{3} – ax^{3}) + (kx^{2} + x^{2}) + (-3.5x – 2x) + (1 + 0.5) =

= x^{4} + (7 – a)x^{3} + (k + 1) x^{2} + (-5.5x) + 1.5 =

= x^{4} + (7 – a)x^{3} + (k + 1) x^{2} – 5.5x + 1.5

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