Algebra Tutorials!

 Thursday 20th of July

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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Polynomials

A term is an expression containing a number or the product of a number and one or more variables raised to powers. Some examples of terms are

4x3, -x2y3, 6ab, and -2.

A polynomial is a single term or a finite sum of terms. The powers of the variables in a polynomial must be positive integers. For example,

4x3 + (-15x2) + x + (-2)

is a polynomial. Because it is simpler to write addition of a negative as subtraction, this polynomial is usually written as

4x3 -15x2 + x -2

The degree of a polynomial in one variable is the highest power of the variable in the polynomial. So 4x3 -15x2 + x -2 has degree 3 and 7w -w2 has degree 2.

The degree of a term is the power of the variable in the term. Because the last term has no variable, its degree is 0.

 4x3 -15x2 + x -2 ↓ ↓ ↓ ↓ Third-degree term Second-degree term First-degree term Zero-degree term

A single number is called a constant and so the last term is the constant term. The degree of a polynomial consisting of a single number such as 8 is 0.

The number preceding the variable in each term is called the coefficient of a variable or the coefficient of that term. In 4x3 -15x2 + x -2 the coefficient of x3 is 4, the coefficient of x2 is -15, and the coefficient of x is 1 because x = 1 Â· x.

Identifying coefficients

Determine the coefficients of x3 and x2 in each polynomial:

a) x3 + 5x2 - 6

b) 4x6 - x3 + x

Solution

a) Write the polynomial as 1 Â· x3 + 5x2 - 6 to see that the coefficient of x3 is 1 and the coefficient of x2 is 5.

b) The x2-term is missing in 4x6 - x3 + x. Because 4x6 - x3 + x can be written as

4x6 - 1 Â· x3 + 0 Â· 4x6 - x2 + x,

the coefficient of x3 is -1 and the coefficient of xis 0.

For simplicity we generally write polynomials with the exponents decreasing from left to right and the constant term last. So we write

x3 - 4x2 + 5x + 1 rather than  - 4x2 + 1 + 5x + x3

When a polynomial is written with decreasing exponents, the coefficient of the first term is called leading coefficient.

Certain polynomials are given special names. 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. For example, 3x5 is a monomial, 2x - 1 is a binomial, and 4x6 -3x + 2 is a trinomial.

Types of polynomials

Identify each polynomial as a monomial, binomial, or trinomial and state its degree.

a) 5x2 - 7x3 + 2

b) x43 - x2

c) 5x

d) -12

Solution

a) The polynomial 5x2 - 7x3 + 2 is a third-degree trinomial.

b) The polynomial x43 - x2 is a binomial with degree 43.

c) Because 5x = 5x1, this polynomial is a monomial with degree 1.

c) The polynomial -12 is a monomial with degree 0.