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3. Polynomial Functions & Expressions
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United States of AmericaFL
Grade 912

3.01 Operations with polynomials

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Concept summary

A polynomial is an expression made up of terms which have variables raised to non-negative integer powers and which have real or complex coefficients.

Coefficient

The number or constant that multiplies a variable in an algebraic expression. If no number is specified, the coefficient is 1.

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. A quadratic expression is an example of a polynomial.

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 can also have special names:

Monomial

A polynomial with only one term

Binomial

A polynomial with two terms

Trinomial

A polynomial with three terms

Perfect square trinomial (PST)

A trinomial that is formed by multiplying a binomial by itself, such as \\x^2 + 2x + 1 = \left(x + 1\right)^2

Worked examples

Example 1

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

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 - 9Distribute the subtraction
\displaystyle =\displaystyle x^2y + 6xy - x^2 - 5xy^2 - 9Combine like terms

Example 2

Fully simplify the polynomial expression \left(3 + 2a\right)^3.

Approach

To simplify this expression we can rewrite the exponent as repeated multiplication, then carefully multiply each term in one factor by each term in the next factor.

Solution

\displaystyle \left(3 + 2a\right)^3\displaystyle =\displaystyle \left(3 + 2a\right)\left(3 + 2a\right)\left(3 + 2a\right)Rewrite the exponent as multiplication
\displaystyle =\displaystyle \left(9 + 12a + 4a^2\right)\left(3 + 2a\right)Multiply the first two factors as a PST
\displaystyle =\displaystyle 27 + 36a + 12a^2 + 18a + 24a^2 + 8a^3Multiply the two remaining factors
\displaystyle =\displaystyle 27 + 54a + 36a^2 + 8a^3Combine like terms

Reflection

Notice that the leading term (the term with the highest power of the variable) for this polynomial is 8a^3, even though this is not the first term that is written.

Example 3

Form a fully simplified polynomial expression for the perimeter of the rectangle shown.

A rectangle with length of a side labelled as 6 x plus 5 y and width 9 x plus 4 y.

Approach

The perimeter of a shape is the sum of its side lengths. In this case, the shape is a rectangle, so we can add the two labeled side lengths and then double the result.

Solution

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 2\left(6x + 5y + 9x + 4y\right)
\displaystyle =\displaystyle 2\left(15x + 9y\right)Combine like terms
\displaystyle =\displaystyle 30x + 18yDistribute the multiplication by 2

Reflection

We could alternatively have doubled each side length first, then added the two results:

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 2\left(6x + 5y\right) + 2\left(9x + 4y\right)
\displaystyle =\displaystyle 12x + 10y + 18x + 8yDistribute each multiplication by 2
\displaystyle =\displaystyle 30x + 18yCombine like terms

Doing so gives the same result for the perimeter.

Outcomes

MA.912.AR.1.3

Add, subtract and multiply polynomial expressions with rational number coefficients.

MA.912.AR.1.6

Solve mathematical and real-world problems involving addition, subtraction, multiplication or division of polynomials.

MA.912.F.3.2

Given a mathematical or real-world context, combine two or more functions, limited to linear, quadratic, exponential and polynomial, using arithmetic operations. When appropriate, include domain restrictions for the new function.

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