5A - Introduction to polynomials

Polynomials

A univariate polynomial is an expression involving different terms with the same variable raised to different powers. In general, a polynomial can be written in the form:

• In this univariate polynomial, the single variable (pronumeral) is x.
• a_n, a_n-1, ... , a₁, a₀ are all real numbers (ℝ). They are the coefficients of the pronumeral (x).
• The index, or powers, are represented by n, n-1, ..., 1, 0 in the equation above. For polynomials the powers are positive integers (ℤ⁺) or zero.

Language and definitions of polynomials

• The leading term of a polynomial contains the variable (x) raised to the highest power (n).
• The degree, or order, of the polynomial is determined by the value of the highest power (index) present.
• A monic polynomial contains the leading term with a coefficent equal to 1. That is, the term with the highest power has a 1 as the coefficient.
• The final term, a₀, is referred to as a constant term, where the index of the variable is zero (x⁰ = 1).
• 0 is known as the zero polynomial.

Common polynomials

There are several common polynomials which we have studied or will study in this course.

• A linear function (such as y = mx + c) is a polynomial of degree 1.
• A quadratic function (such as y = ax² + bx + c) is a polynomial of degree 2.
• A cubic function (such as y = ax³ + bx² + cx + d) is a polynomial of degree 3.
• A quartic function (such as y = ax⁴ + bx³ + cx² + dx + e) is a polynomial of degree 4.

Arithmetic involving polynomials

It is possible to perform addition, subtraction, multiplication and division on polynomials, when we do so the answer will result in another polynomial. The degree of a polynomial is represented as deg(f). For two polynomials, f and g, the following hold true:

• deg(f + g) ≤ max{deg( f ), deg(g)}
• deg(f × g) = deg( f ) + deg(g)

5A - VIDEO EXAMPLE 1:

Consider the polynomials P(x) = 3x³ + 4x - 3 and Q(x) = x³ - 7x, find:

• P(x) + Q(x)
• P(x) - Q(x)
• P(x) × Q(x)

Equating coefficients

Two polynomials are equal when they give the same same value for all values of the variable (x). It can be shown that, two polynomials are equal if and only if their corresponding coefficients are equal. This property gives rise to the process of equating coefficients when working with polynomials.

5A - VIDEO EXAMPLE 2:

If y = ax³ + bx² + cx + d is equal to P(x) = 2x³ + 4x - 3 state the value of a, b, c and d.

5A - VIDEO EXAMPLE 3:

The polynomial Q(x) = 2x³ - 6x² + 6x + 2 can be written in the form A(x - h)³+ k, where A,

h and k are real numbers. Find the values of A, h and k.