Operations Using Polynomials: Workbook Examples/Practice

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5 Operations Using Polynomials: Workbook Examples/Practice

6.4 Basic Operations Using Polynomials

Many applications in mathematics have to do with what are called polynomials. Polynomials are made up of terms. Terms are a product of numbers and/or variables. For example, $5x$ , $2y^2$ , $-5$ , $ab^3c$ , and $x$ are all terms. Terms are connected to each other by addition or subtraction.

Expressions are often defined by the number of terms they have.

$\begin{array}{l} \text{A monomial has one term, such as } x, xy, 3x^2.\\ \text{A binomial has two terms connected by a + or -, such as } a^2-b^2, 3x-y, 4x+2xy^3 \\ \text{A trinomial has three terms connected by a + or -, such as } ax^2 + bx + c \end{array}$

If it is known what the variable in a polynomial represents, it is possible to substitute in the value and evaluate the polynomial, as shown in the following example.

Example 1: Simplify the expression $2x^2 - 4x + 6$ using $x = -2$ .

Remember the exponent only affects the number to which it is physically attached. This means −3² = −9 because the exponent is only attached to the 3, whereas (−3)² = 9 because the exponent is attached to the parentheses and everything inside.

Example 2: Simplify the expression $-x^2 + 2x + 6$ using $x = 4$ .

Sometimes, when working with polynomials, the value of the variable is unknown and the polynomial will be simplified rather than ending with some value. The simplest operation with polynomials is addition. When adding polynomials, it is merely combining like terms. Consider the following example.

Example 3: Simplify the expression $(2x^3 - 4x + 8) + (3x^3 - 10x^2 - 20).$

Subtracting polynomials is almost as fast as addition. Subtraction is generally shown by a minus sign in front of the parentheses. When there is a negative sign in front of parentheses, distribute it throughout the expression, changing the signs of everything inside.

Example 4: Simplify the expression $(8x^2 - 4x + 7) - (5x^2 + 6x - 3).$

Examples/Practice

Simplify each expression using the variables given.

1.) $-a^3 - a^2 + 6a - 21 \text{ when }a = -4$

2.) $n^2 + 3n - 11 \text{ when }n = -6$

3.) $-5n^4 - 11n^3 - 9n^2 - n - 5 \text{ when } n = -1$

4.) $x^4 - 5x^3 - x + 13 \text{ when } x = 5$

5.) $(6x^3 + 5x) - (8x + 6x^3)$

6.) $(5n^4 + 6n^3) + (8 - 3n^3 - 5n^4)$

7.) $(8x^2 + 1) - (6 - x^2 - x^4)$

8.) $(2a + 2a^4) - (3a^2 - 5a^4 + 4a)$

6.5 Multiplication of Polynomials

Multiplying monomials is done by multiplying the numbers or coefficients and then adding the exponents on like factors. This is shown in the next example.

Example 1: Find the following product: $(3x^3y^4z)(2x^2y^6z^2).$

Some notes: $z$ has an exponent of 1 when no exponent is written. When adding or subtracting, the exponents will stay the same, but when multiplying (or dividing), the exponents will change.

Next, consider multiplying a monomial by a polynomial. We have seen this operation before with distributing throughout parentheses. Here, its the exact same process.

Example 2: Find the following product: $2x^3 (3x^2 - 2x + 5).$