# 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) + 4x^{2} − x^{3}

2. Simplify the given expression.

(5x^{2} − 3y^{2}) + (3x^{2} − 2xy + 6y^{2})

3. Perform the indicated operation .

Subtract 3m^{3} + 5m + 5 from 4m^{3} − 5m − 5

4. Perform the indicated subtraction.

(−9t^{4} + 7t^{2} − 1) − (−9t^{4} + 8t^{2} + 10)

**2 Concepts**

When we look at an expression like 5x^{4}+2x^{3}−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, 5x^{4} 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 3x^{4}−2x^{3}+x^{2}−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.

2x^{2} + 11 − 5x^{3} − x

Answer. The degree of the polynomial is 3;
−5x^{3}+2x^{2}−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:

(4x^{3} + 2x^{2} + x − 10) + (2x^{2} − 7x+4) = 4x^{3} + 4x^{2} − 6x
− 6

In the next example, we subtract two polynomials.

**3.2 Example**

Subtract:

8x^{2} − 4x + 7 from 9x^{3} − 11x^{2} + 10x − 3

We put the expression after the word from first.

(9x^{3} − 11x^{2} + 10x − 3) − (8x^{2} − 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.

(9x^{3} − 11x^{2} + 10x − 3) + (−8x^{2} + 4x − 7)

=

9x^{3} − 19x^{2} + 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) + 4x^{2} − x^{3}

2. Simplify the given expression.

(5x^{2} − 3y^{2}) + (3x^{2} − 2xy + 6y^{2})

3. Subtract

3m^{3} + 5m + 5 from 4m^{3} − 5m − 5

4. Perform the indicated substraction.

(−9t^{4} + 7t^{2} − 1) − (−9t^{4} + 8t^{2} + 10)

1. Identify which is a monomial, binomial, trinomial or
other type of polynomial.

What is the degree of the polynomial?

(1 − 2x) + 4x^{2} − x^{3}

This is a polynomial of degree 3.

2. Simplify the given expression.

(5x^{2} − 3y^{2}) + (3x^{2} − 2xy + 6y^{2})

=

8x^{2} − 2xy + 3y^{2}

3. Subtract

3m^{3} + 5m + 5 from 4m^{3} − 5m − 5

4^{m3} − 5m − 5 − (3m^{3} + 5m + 5)

4m^{3} − 5m − 5 + (−3m^{3} − 5m − 5)

=

m^{3} − 10m − 10

4. Perform the indicated substraction.

(−9t^{4} + 7t^{2} − 1) − (−9t^{4} + 8t^{2} + 10)

=

(−9t^{4} + 7t^{2} − 1) + (9t^{4} − 8t^{2} − 10)

=

−t^{2} − 11

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