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Polynomials

1 Polynomials 5.2

By the end of this section, you should be able to solve the following problems.

1. Identify the following as a monomial , binomial, trinomial, or other type
of polynomial. What is the degree of the expression?

(1 − 2x) + 4x2 − x3

2. Simplify the given expression.

(5x2 − 3y2) + (3x2 − 2xy + 6y2)

3. Perform the indicated operation .

Subtract 3m3 + 5m + 5 from 4m3 − 5m − 5

4. Perform the indicated subtraction.

(−9t4 + 7t2 − 1) − (−9t4 + 8t2 + 10)

2 Concepts

When we look at an expression like 5x4+2x3−7x+4, what we see are variables
separated by plus and minus signs. Each one of those variable parts is called
a term. Within each term the number part is called the coefficient and the
exponent is called the degree of the term. For instance, 5x4 has a coefficient
of 5 and a degree of 4. Taken by itself, a single term is called a monomial.
Two terms added together are called a binomial. A Three term expression
is called a trinomial. More than three terms is simply called a polynomial.
In any polynomial, the term of highest degree determines the degree of the
polynomial. So we would say that 3x4−2x3+x2−7 is a polynomial of degree
4. Note that it is customary to list the terms in a polynomial in order of
degree
from highest to lowest from left to right.

2.1 Example

State the degree of the polynomial, and arrange the terms in order of degree
from left to right. List the coefficients of each term, including the constant
term, in roster form.

2x2 + 11 − 5x3 − x

Answer. The degree of the polynomial is 3; −5x3+2x2−x+11; {−5, 2,−1, 11}.

3 Concepts

When we add two polynomials together, we add them term by term and like
terms to like terms . Similarly we subtract polynomials by subtracting them
term by term only subtracting like terms . In the next example, we add two
polynomials.

3.1 Example

Add:

(4x3 + 2x2 + x − 10) + (2x2 − 7x+4) = 4x3 + 4x2 − 6x − 6

In the next example, we subtract two polynomials.

3.2 Example

Subtract:

8x2 − 4x + 7 from 9x3 − 11x2 + 10x − 3

We put the expression after the word from first.

(9x3 − 11x2 + 10x − 3) − (8x2 − 4x + 7)

Next, we change the subtraction sign to addition and change all the signs
in the expression to the right of the subtraction sign to their opposites . Then
we add like terms.

(9x3 − 11x2 + 10x − 3) + (−8x2 + 4x − 7)

=

9x3 − 19x2 + 14x − 10

4 Facts

1. The coefficient of a term is the large number next to the variable.

2. The degree of a polynomial is the degree of term that has the highest
power in the polynomial .

3. When adding or subtracting polynomials, we add or subtract like terms
term by term.

4. If a problem is written: Subtract a from b, we rewrite it to say b − a.

5. When we subtract in algebra , we add the opposite.

5 Exercises

1. Identify which is a monomial, binomial, trinomial or other type of polynomial.
What is the degree of the polynomial?

(1 − 2x) + 4x2 − x3

2. Simplify the given expression.

(5x2 − 3y2) + (3x2 − 2xy + 6y2)

3. Subtract

3m3 + 5m + 5 from 4m3 − 5m − 5

4. Perform the indicated substraction.

(−9t4 + 7t2 − 1) − (−9t4 + 8t2 + 10)

6 Solutions

1. Identify which is a monomial, binomial, trinomial or other type of polynomial.
What is the degree of the polynomial?

(1 − 2x) + 4x2 − x3

This is a polynomial of degree 3.

2. Simplify the given expression.

(5x2 − 3y2) + (3x2 − 2xy + 6y2)

=

8x2 − 2xy + 3y2

3. Subtract

3m3 + 5m + 5 from 4m3 − 5m − 5

4m3 − 5m − 5 − (3m3 + 5m + 5)

4m3 − 5m − 5 + (−3m3 − 5m − 5)

=

m3 − 10m − 10

4. Perform the indicated substraction.

(−9t4 + 7t2 − 1) − (−9t4 + 8t2 + 10)

=

(−9t4 + 7t2 − 1) + (9t4 − 8t2 − 10)

=

−t2 − 11
 

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