Identify terms and coefficients.

Know the vocabulary for polynomials.

Add like terms.

Add and subtract polynomials.

Evaluate polynomials.

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21

Unit – 3.3 - Polynomials

Identify terms and coefficients.

In an expression such as

the quantities 4x3, 6x2, 5x, and 8 are called terms. In the first term

4x3, the number 4 is called the coefficient, of x3. In the same way, 6

is the coefficient of x2 in the term 6x2, and 5 is the coefficient of x in

the term 5x. The constant term is 8 .

3 24 ,6 5 8x x x

Slide 5.4-4

Solution:

2, 1

Name the coefficient of each term in the expression32 .x x

Slide 5.4-5

EXAMPLE 1 Identifying Coefficients

Know the vocabulary for polynomials.

A polynomial in x is a term or the sum of a finite number of terms of the form axn, for any real number a and any whole number (no negative, no fraction) n. For example,

is a polynomial in x. (The 4 can be written as 4x0.) This polynomial is written in standard form, since the exponents on x decrease from left to right.

By contrast,

is not a polynomial in x, since x appears in a denominator.

8 6 4 216 7 5 3 4x x x x

3 2 12x x

x

Polynomial in x

Not a Polynomial

Slide 5.4-10

A polynomial can be defined using any variable and not just x. In fact, polynomials may have terms with more than one variable.

The degree of a term is the sum of the exponents on the variables. For example 3x4 has degree 4, while the term 5x (or 5x1) has degree 1, −7 has degree 0 ( since −7 can be written −7x0).

The degree of a polynomial is the greatest degree term of the polynomial. For example 3x4 + 5x2 + 6 is of degree 4.

Slide 5.4-11

Know the vocabulary for polynomials. (cont’d)

Three types of polynomials are common and given special names. A polynomial with only one term is called a monomial. (Mono- means “one,” as in monorail.) Examples are

6.9 ,m 5 ,6y 2 ,a andMonomials

A polynomial with exactly two terms is called a binomial. (Bi- means “two,” as in bicycle.) Examples are

4 39 9 ,x x 28 6 ,m m 5 23 .9m mand

Binomials

A polynomial with exactly three terms is called a trinomial. (Tri- means “three,” as in triangle.) Examples are

3 2 6,9 4m m 219 8,5

3 3y y 5 23 9 .2m m and

TrinomialsSlide 5.4-12

Know the vocabulary for polynomials. (cont’d)

Degree Name Example

0 Constant 5

1 Linear 2xy + 4

2 Quadratic 4x2 – 7x + 2

3 Cubic x3 – 2x2 – 2x + 1

4 Quartic -8x4 – 7x2 + 3x - 4

5 Quintic 3x5 – 5x4 + 2x3 – 4x2 +10y

Number of Terms Name Example

1 Monomial 3x2

2 Binomial 5x + 4x3

3 Trinomial 3x + 4x3 - 7

More than 3 Polynomial 5x5 + 4x4 - 2x3 + 8x2 - x - 1

Know the vocabulary for polynomials. (cont’d)

Solution:

Write polynomial in standard form, give the degree, and tell whether the polynomial is a monomial, binomial, trinomial.

Slide 5.4-13

EXAMPLE 3 Classifying Polynomials

3x + 5x3 - 4

5x3 + 3x – 4

Degree 3 or cubic, trinomial

Add like terms.

like terms have exactly the same combinations of variables, with the same exponents on the variables. Only the coefficients may differ.

We combine, or add, like terms by adding their coefficients.

3 3and 19 14m m

Examples of like terms

9 9 9, , and 6 37y y yand 2 3pq pq

2 2and 2xy xy

Slide 5.4-7

Solution:

2 23 5r r r

Simplify by adding like terms.

26 3r r

Unlike terms cannot be combined. Unlike terms have different variables or different exponents on the same variables.

Slide 5.4-8

EXAMPLE 2 Adding Like Terms

Add and subtract polynomials.

Polynomials may be added, subtracted, multiplied, and divided.

Subtracting PolynomialsTo subtract two polynomials, change all the signs in the second polynomial and add the result to the first polynomial.

Adding PolynomialsTo add two polynomials, add like terms.

Slide 5.4-17

Add.

and

and

Solution:

3 24 3 2x x x 2 2 5x x

3 26 2 3x x x

24 2x

3 24 3 2x x x 3 26 2 3x x x +

2 2 5x x 24x 2+

3 210x x x 25 2 3x x

Slide 5.4-18

EXAMPLE 5 Adding Polynomials Vertically

Solution:

Add.

4 2 4 22 6 7 3 5 2x x x x

4 2 9x x

Slide 5.4-19

EXAMPLE 6 Adding Polynomials Horizontally

Perform the subtractions.

from

Solution:

3 214 6 2 5 .y y y

2 27 11 8 3 4 6y y y y

2 27 11 8 3 4 6y y y y 210 15 2y y

3 2 3 214 6 2 5 2 7 4 6y y y y y y 3 212 6 11y y y

3 22 7 4 6y y y

Slide 5.4-20

EXAMPLE 7 Subtracting Polynomials Horizontally

Subtract.

Solution:

3 214 6 2y y y 3 22 7y y 6

3 214 6 2y y y 3 22 7y y + 63 212 2 6y y y

Slide 5.4-21

EXAMPLE 8 Subtracting Polynomials Vertically

Subtract.

3 2 2 3 2 25 3 4 7 6m n m n mn m n m n mn

Solution:

3 2 2 3 2 25 3 4 7 6m n m n mn m n m n mn

3 2 22 4 10m n m n mn

Slide 5.4-22

EXAMPLE 9 Adding and Subtracting Polynomials with More Than One Variable

Solution:

Find the value of 2y3 + 8y − 6 when y = −1.

312 61 8

2 1 8 6 2 8 6 16

Use parentheses around the numbers that are being substituted for the variable, particularly when substituting a negative number for a variable that is raised to a power, or a sign error may result.

Slide 5.4-15

EXAMPLE 4 Evaluating a Polynomial