Polynomials

Polynomials• A polynomial is a term or the sum or

difference of two or more terms.• A polynomial has no variables in the

denominator.• The “degree of a term” is the exponent of

the variable (4x3 is a 3rd degree term).• The “degree of the polynomial” is the

same as the degree of the term with the highest degree.

(x5 + 4x3 – 3x + 2 is a fifth degree polynomial)

• Polynomials in standard form are in order of degree from highest to lowest with the constant at the end.

Polynomials By Terms

• A polynomial with one term is called a monomial.

• A polynomial with two terms is called a binomial.

• A polynomial with three terms is called a trinomial.

Polynomials by Degrees

• A first degree polynomial is linear.• A second degree polynomial is

quadratic.• A third degree polynomial is cubic.• A polynomial with no variable is

called a constant.

Examples 1 and 2• Name the degree of each term and each

polynomial. Put them in standard form.

• Degree of each term 5, 3, 1, and 0• Degree of the polynomial 5• It’s in standard form.• New Problem:

• Degree of each term 1, 2, 0, and 3• Degree of the polynomial 3rd

• In standard form, it is:

x x x5 34 3 2

5 2 3 92 3x x x

9 2 5 33 2x x x

Model Polynomial Addition and Subtraction

Algebra Tiles

1 unit or -1 unit

x units or-x units

x2 units or-x2 units

Model the following with Algebra Tiles• (2x2 – x) + (x2 + 3x – 1)

3x2 + 2x - 1

(2x2 + 6) – 4x2

-2x2 + 6

What is the expression modeled below?

(2x2 – 2x – 4) + (-x2 + 3x + 2)

Adding Polynomials

• Collect like terms. • In order to have like terms,

the variable parts must be exactly the same.

• Combine the coefficients (the numbers in front of the variable).

Subtracting Polynomials

• Drop the first set of parentheses.

• Distribute a –1 in the second set of parentheses.

• Combine like terms.

To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial:

–(2x3 – 3x + 7)= –2x3 + 3x – 7

Subtract

• (3x2 + 2x – 1) – (x2 + 4x – 2)• Distribute the -1• 3x2 + 2x – 1 - x2 - 4x + 2• Combine like terms

2x2 – 2x + 1

An egg is thrown off the top of a building. Its height in meters above the ground can be approximated by the polynomial 300 + 2t – 4.9t2, where t is the time since it was thrown in seconds.

How high is the egg above the ground after 5 seconds?

How high is the egg above the ground after 6 seconds?

Example 3

187.5

135.6

Example 4

• A firework is launched from a platform 6 feet above the ground at a speed of 200 feet per second. The firework has a 5-second fuse. The height of the firework in feet is given by the polynomial -16t2 + 200t + 6, where t is the time in seconds. How high will the firework be when it explodes?

606

Try these…

Find the degree of each polynomial.

1. 7a3b2 – 2a4 + 4b – 15

2. 25x2 – 3x4

Write each polynomial in standard form. Then

give the leading coefficient.

3. 24g3 + 10 + 7g5 – g2

4. 14 – x4 + 3x2

4

5

–x4 + 3x2 + 14; –1

7g5 + 24g3 – g2 + 10; 7

Try these…

Classify each polynomial according to its degree and number of terms.

5. 18x2 – 12x + 5 quadratic trinomial

6. 2x4 – 1 quartic binomial

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

Try these…

• (5x2 – 2x + 3) + (6x2 + 5x + 6)

• (3x2 – x + 3) + (2x2 + 2x + 9)

• (11x2 + 3x – 1) – ( 2x2 + 2x + 8)

• (x2 – x + 2) – (-3x2 – x – 4)

11x2+3x+9

5x2+x+12

9x2+x-9

5x2+6