Algebra: Polynomials by Stephen Bucaro

Author:Stephen Bucaro [Bucaro, Stephen]
Language: eng
Format: epub
Publisher: bucarotechelp.com
Published: 2020-08-01T00:00:00+00:00

Scientific Notation Answers

CONTENTS

Introduction to Polynomials

CONTENTS

Many applications in mathematics have to do with polynomials. Polynomials are made up of terms. Terms are a product of numbers and/or variables. For example, 5x, 2y2, - 5, ab3c, and x are all terms. Terms are connected to each other by addition or subtraction.

Expressions are often named based on the number of terms in them. A monomial has one term, such as 3x2. A binomial has two terms, such as a2 - b2. A Trinomial has three terms, such as ax2 + bx + c. The term polynomial means many terms. Monomials, binomials, trinomials, and expressions with more terms are all polynomials.

Example 230

2x2 - 4x +6 when x = -4

Replace variable x with -4

2(-4)2 - 4(-4) + 6

Exponents first

2(16) - 4(-4) + 6

Multiplication (we can do all terms at once)

32 + 16 + 6

Add

Solution: 54

It is important to be careful with negative variables and exponents. Remember the exponent only effects the number it is physically attached to. This means -32 = -9 because the exponent is only attached to the 3. Also, (-3)2 = 9 because the exponent is attached to the parenthesis and effects everything inside. When we replace a variable with parenthesis like in the previous example, the substituted value is in parenthesis. So the (-4)2 = 16 in the example. However, consider the next example.

Example 231

-x2 + 2x +6 when x = 3

Replace variable x with 3

-(3)2 + 2(3) + 6

Exponent only on the 3, not negative.

-9 + 2(3) + 6

Multiply

-9 + 6 + 6

Solution: 3

Generally when working with polynomials we do not know the value of the variable, so we will try and simplify instead. The simplest operation with polynomials is addition. When adding polynomials we are mearly combining like terms. Consider the following example. Example 232

(4x3 - 2x + 8) + (3x3 - 9x2 - 11)

Combine like terms 4x3 + 3x3 and 8 - 11

Solution: 7x3 - 9x2 - 2x - 3

Generally final answers for polynomials are written so the exponent on the variable counts down. Example 233 demonstrates this with the exponent counting down 3, 2, 1, 0 (recall x0 = 1). Subtracting polynomials is almost as fast. One extra step comes from the minus in front of the parenthesis. When we have a negative in front of parenthesis we distribute it through, changing the signs of everything inside. The same is done for the subtraction sign.

Example 233

(5x2 - 2x +7) - (3x2 + 6x - 4)

Distribute negative through second part.

5x2 - 2x +7 - 3x2 - 6x + 4

Combine like terms 5x2 - 3x3, -2x - 6x, and 7 + 4

Solution: 2x2 - 8x + 11

Addition and subtraction can also be combined into the same problem as shown in this final example.

Example 234

(2x2 - 4x + 3) + (5x2 - 6x + 1) - (x2 - 9x + 8)

Distribute negative through

2x2 - 4x + 3 + 5x2 - 6x + 1 - x2 + 9x - 8

Combine like terms

Solution: 6x2 - x - 4

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