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# Polynomials And Polynomial Functions

Categories: Math

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 – 4x² + 7x − 8

• is a sum of three terms: 4x2, 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 4x² + 7x − 8 has the value 3. If  x has the value 0, then 4x² + 7x − 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.

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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 axn, where a is a real number and n is a whole number. Therefore, we can conclude that expression 4x² + 7x − 8 is polynomial in x. This polynomial has three terms.

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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 4x² is 2, the degree of 7x 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 4x² + 7x − 8 is 2.

Give an example to show how to:

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

1. 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) = 4x2 – 12x – 2x + 4 – 3 + x + x2 =

= 4x2 + x2 – 12x – 2x + x + 4 – 3

4x2 and x2 are like terms;
-12x, -2x, and x are like terms;

4 and -3 are like terms.

4x2 + x2 – 12x – 2x + x + 4 – 3 = (4x2 + x2) + (-12x – 2x + x) + (4 – 3) = 5x2 – 13x + 1.

1. First polynomial:x4 + 7x3 + kx2 – 3.5x + 1

Second polynomial:   ax3 – x2 + 2x – 0.5

(x4 + 7x3 + kx2 – 3.5x + 1) + (ax3 – x2 + 2x – 0.5) =

= x4 + 7x3 + kx2 – 3.5x + 1 + ax3 – x2 + 2x – 0.5 =

= x4 + 7x3 + ax3 + kx2 – x2 – 3.5x + 2x + 1 – 0.5 =

= x4 + (7x3 + ax3) + (kx2 – x2) + (-3.5x + 2x) + (1 – 0.5) =

= x4 + (7 + a)x3 + (k – 1) x2 + (-1.5x) + 0.5 =

= x4 + (7 + a)x3 + (k – 1) x2 – 1.5x + 0.5

Subtracting:

(x4 + 7x3 + kx2 – 3.5x + 1) – (ax3 – x2 + 2x – 0.5) =

= x4 + 7x3 + kx2 – 3.5x + 1 – ax3 + x2 – 2x + 0.5 =

= x4 + 7x3 – ax3 + kx2 + x2 – 3.5x – 2x + 1 + 0.5 =

= x4 + (7x3 – ax3) + (kx2 + x2) + (-3.5x – 2x) + (1 + 0.5) =

= x4 + (7 – a)x3 + (k + 1) x2 + (-5.5x) + 1.5 =

= x4 + (7 – a)x3 + (k + 1) x2 – 5.5x + 1.5