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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) + 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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