For example, for the polynomial expression 2x^2+\left(\sqrt{3}+1\right)x-\dfrac{3}{4}, the coefficient of the quadratic term is an integer: 2, the coefficient of the linear term is irrational: \sqrt{3}+1, and the constant is rational: -\dfrac{3}{4}.

A polynomial in one variable is of the form a_nx^n + a_{n - 1}x^{n - 1} + \ldots + a_1x + a_0, where n is a non-negative integer. Linear expressions, quadratic expressions, and cubic expressions are all examples of polynomials.

The term which has a fixed value and no variables is called the constant term. The term with highest exponent on the variable is called the leading term, and the exponent of this term is the degree of the polynomial.

a polynomial expression labelled with its parts written as: p of x is equal to a sub n times x to the nth power plus a sub n - 1 times x to the power of n - 1 plus ellipsis plus a sub 2 times x squared plus a sub 1 times x plus a sub 0. a sub n times x to the nth power is the leading term, a sub n is the leading coefficient and the power n is the degree. In the term a sub n - 1 times x to the power of n - 1, a sub n - 1 is the coefficient. The term a sub 2 times x squared is the quadratic term, a sub 1 times x is the linear term and a sub 0 is the constant term.

Polynomials may be in more than one variable, such as 3xy^3+2xy+11y+3. In this case, the degree of a term will be the sum of the exponents for all variables. For 3xy^3+2xy+11y+3, the term 3xy^3 has degree 4, so the polynomial also has degree 4.

Polynomials can also have special names:

Monomial

A polynomial with only one term

Example:

5xy

Binomial

A polynomial with two terms

Example:

2x^3-4

Trinomial

A polynomial with three terms

Example:

8x^2-5x+3

Worked examples

Example 1

Consider the polynomial expression \left(x^2y + 4xy - x^2\right) - \left(5xy^2 - 2xy + 9\right).

Fully simplify the polynomial expression.

Approach

We can simplify this expression by combining like terms. To do so, we must be careful to apply the subtraction to each term in the second polynomial. Also remember that like terms must have the same variables with the same exponents.

Solution

\displaystyle \left(x^2y + 4xy - x^2\right) - \left(5xy^2 - 2xy + 9\right)	\displaystyle =	\displaystyle x^2y + 4xy - x^2 - 5xy^2 + 2xy - 9	Distribute the subtraction
	\displaystyle =	\displaystyle x^2y + 6xy - x^2 - 5xy^2 - 9	Combine like terms

State the degree of the simplified polynomial.

Approach

The degree is the maximum of the sum of the exponents for each term.

Solution

Both the terms x^2y and -5xy^2 have degree 3, so this polynomial has degree 3.

Reflection

For polynomials in more than one variable, there may be more than one way to define the leading term or leading coefficient. It depends on how you define the monomial order - as an extension see what you can learn about monomial order.

Example 2

The balance of a savings account in dollars is represented by the polynomial 50+6x, where x is the number of weeks since the account was opened.

Interpret the constant in this polynomial.

Approach

The constant is not affected by the number of weeks which have passed as it does not have a variable part.

Solution

The constant represents the initial amount deposited into the savings account.

Reflection

This model is assuming no interest is earned. Sometimes banks have a minimum balance required to earn interest.

Interpret the linear term in this polynomial, including its coefficient.

Approach

The linear term has a variable component and as x increases by 1, 6x would increase by 6.

Solution

6x represents the amount deposited into the account through weekly deposits. In particular, this person is depositing \$6 per week.

Example 3

A rectangular swimming pool is 16\text{ yds} long and 6\text{ yds} wide. It is surrounded by a pebble path of uniform width x\text{ yds}.

A rectangular swimming pool with length 16 yards, and width 6 yards. A path forms a rectangle around the pool x yards in width from each side of the pool.

Find an expression for the area of the path in terms of x. Fully simplify your answer.

Approach

The area of the path will be the area of the larger rectangle minus the area of the pool.

Solution

\displaystyle A_{\text{path}}	\displaystyle =	\displaystyle A_{\text{Large rectangle}}-A_{\text{pool}}
	\displaystyle =	\displaystyle \left(16+2x\right)\left(6+2x\right)-\left(16\right)\left(6\right)
	\displaystyle =	\displaystyle 96 +32x+12x+4x^2-96
	\displaystyle =	\displaystyle 4x^2+44x

A_{\text{path}}=4 x^{2} + 44 x\text{ yd}^2

Outcomes

M3.A.SSE.A.1

Interpret expressions that represent a quantity in terms of its context.*

M3.A.SSE.A.1.A

Interpret parts of an expression, such as terms, factors, and coefficients.

M3.A.SSE.A.1.B

Interpret complicated expressions by viewing one or more of their parts as a single entity.

1.02 Polynomial expressions

Concept summary

Worked examples

Example 1

Approach

Solution

Approach

Solution

Reflection

Example 2

Approach

Solution

Reflection

Approach

Solution

Example 3

Approach

Solution

Outcomes

M3.A.SSE.A.1

M3.A.SSE.A.1.A

M3.A.SSE.A.1.B

M3.MP1

M3.MP3

M3.MP6

M3.MP8

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