Transcript Objective - To simplify polynomials.

Lesson 7.6
Polynomials
California
Standards
Preparation for 10.0 Students add, subtract,
multiply, and divide monomials and
polynomials. Student solve multistep
problems, including word problems, by using
these techniques.
Words to Know
Monomial- Single term consisting of numbers,
single
variables, and whole number exponents.
7x , 3m , 10ab c , 2.4y
2
2
Polynomial- An expression consisting of two or more
many
monomial terms that can’t be combined.
Types of Polynomials
Binomials - Two terms
2x  5 , 3k  1 , 8x  x
2
4
3
Trinomials - Three terms
2
4
2
5x  x  4 , 7t 10t  t
2
Identify each expression as a monomial, binomial,
trinomial, or polynomial. (More than one may apply.)
1) 4x  6
4) 8y  2y  3y  4
3
Binomial, Polynomial
2) 18x  6x  7x
3
2
Trinomial, Polynomial
3) 9t
4
Monomial
2
Polynomial
5) 4t  3
2
Binomial, Polynomial
6) 7x  x 1
2
Trinomial, Polynomial
Standard Form of the Polynomial
- Powers of the variable written in decreasing order.
Non-standard Form
2
5
4
8  4x  7x  3x  x
Standard Form
4
2
5
7x  x  4x 3x  8
You try!
Write the polynomials below in standard form.
1) y  5y  2
4
5y  y  2
3) 8t  t  2  4t  t
6
3
2
 t  4t  8t  t  2
4
2
3
2) 6  m  2m  m
4) k  3  2k  9k
2m  m  m  6
2k  k  9k  3
3
5
5
3
3
4
4
3
2
6
2
The degree of a polynomial is the degree of the term
with the greatest degree.
When written in standard form, the coefficient of the
first term is called the leading coefficient.
Write the polynomial in standard form. Then
give the leading coefficient.
6x – 7x5 + 4x2 + 9
Find the degree of each term. Then arrange
them in descending order:
6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9
Degree 1
5
2
0
5
2
The standard form is–7x5 + 4x2 + 6x + 9.
The leadingcoefficient is –7.
1
0
Remember!
A variable written without a coefficient has a
coefficient of 1.
5
5
y = 1y
Now you try…
Write the polynomial in standard form. Then
give the leading coefficient.
18y5 – 3y8 + 14y
Find the degree of each term. Then arrange
them in descending order:
18y5 – 3y8 + 14y
–3y8 + 18y5 + 14y
Degree
5
8
1
8
The standard form is–3y8 + 18y5 + 14y.
The leading coefficient is –3.
5
1
Simplifying Polynomials
- Simplified by combining like terms.
3x  2x  7x  5x  3x  9
4
2
4
2
10x  x  5x  9
Simplify the polynomials below. Write them in
standard form.
2
5
2
5
1) 6t  t  7t  5t  4t  6  t
2
4
5t  11t  6t  6
2
5
2)  4a  9a  6a  a  3a  a
3
2
3
5a  6a  7a
3
2
Simplify the following polynomials. Write in
standard form.
1) 3m  4m  m  6  7m
2
2
4m  3m 2  6
2
3m  4m  6
2) 5  10t  2t  8t  6  9t
2
2
11  2t  7t 2
7t  2t  11
3
3
3) y  9y  2  8y  y  11
2
9y  8y  9
3
8y  9y  9
3
Simplify the following polynomials. Write in
standard form.
Answers
1)t  t  5t  2t
2
2
2)2  6 x  5 x  7
3
3
3)2n  1  12n  8
4) y  2 y  3 y
2
5)8  4 x  10x  11
2
2
1)3t 2  4t
2)  x 3  5
3)14n  7
4)2 y 2  2 y
5)6 x 2  3
Lesson Quiz: Part I
Find the degree of each polynomial.
1. 7a3b2 – 2a4 + 4b – 15 5
2. 25x2 – 3x4
4
Write each polynomial in standard form.
Then give the leading coefficient.
3. 24g3 + 10 + 7g5 – g2
4. 14 –
x4
+
3x2
7g5 + 24g3 – g2 + 10; 7
–x4 + 3x2 + 14; –1
Lesson Quiz: Part II
Tell whether each number is a root of
3p2 – 8 + 4.
5. 2 yes
6. –2
no
Lesson Quiz: Part III
7. The polynomial 3.675v + 0.096v2 is used to
estimate the stopping distance in feet for a
car whose speed is v miles per hour on flat,
dry pavement. What is the stopping distance
for a car traveling at 70 miles per hour?
727.65 ft