19 20 Polynomials

Question 1

What is a polynomial?

Answer 1

An expression in the form:

a₀ + a₁x + a₂x² + … + a_nxⁿ

^{(listed in reverse order in its standard form)}

a is just any number

all exponents are positive integers

Question 2

Can √5 be a coefficient?

An exponent?

Answer 2

√5 can be a coefficient.

√5 cannot be an exponent (it is not an integer)

Question 3

Can -2 be a coefficient?

An exponent?

Answer 3

• 2 can be a coefficient
• 2 cannot be an exponent because it is not a positive integer

Question 4

Can √x be in a polynomial?

Answer 4

No.

Because √x = x^½ and exponents must be positive integers

Question 5

Can ^{(x² + 2x)}/_{(x³ + 1)} be a polynomial?

Answer 5

No.

The x³ in the denominator is x^⅓ which is not a positive integer.

Question 6

What is the degree of a polynomial?

What are the names of the first 5 polynomial degrees?

Answer 6

The highest exponent.

linear

quadratic

cubic

quartic

quintic

Question 7

What is the leading coefficient of a polynomial?

Answer 7

The coefficient of the highest degree polynomial

Question 8

How do you classify polynomials by quantity of terms?

Answer 8

monomial → 3x²

binomial → 3x² + 2

trinomial → 3x² + 2x + 1

Question 9

Classify 5x³ -1

Answer 9

a cubic binomial

Question 10

Classify A(x) = πx²

Answer 10

a quadratic monomial

Question 11

What is the domain of all polynomials?

Answer 11

AR#s

Question 12

What does the range of polynomials depend on?

Answer 12

The degree.

If odd, the range is AR#s.

If even, the range is not AR#s because both ends will either go up or down so the range is restricted.

Question 13

Can the graph of a polynomial have a sharp corner?

Answer 13

No. That’s why x^⅔ is not a polynomial.

It has a sharp corner at the origin.

Question 14

Can the graph of a polynomial have a break?

Answer 14

No. Because the domain is AR#s.

Question 15

What is the maximum number of x-intercepts a polynomial can have?

Answer 15

the degree of the polynomial

Question 16

What happens to the ends of the graph of a polynomial?

Answer 16

They never flatten out.

If it is an even polynomial, they point in the same direction.

If odd, they point in opposite directions.

Question 17

In an even coefficient, how do you know if the ends point up or down?

Answer 17

If the lead coefficient is positive, they point up.

^{This is because negative xs give the same y as the corresponding positive x}

Question 18

In an odd coefficient, how do you know if which end points up?

Answer 18

If the lead coefficient is positive, the left side points down and the right side points up.

^{In other words, the graph is always going up}

Question 19

Why does the lead coefficient control the direction of the ends?

Answer 19

It is the dominant term (has greatest effect)

Question 20

For f(x) = x³ :

classify

how many x-intercepts?

ends up or down?

Answer 20

It is a cubic monomial.

1 x-intercept (0,0)

points down on the left, up on the right

Question 21

How many turns does the graph of a cubic polynomial have?

Answer 21

A cubic polynomial can have, at most, 3 roots, so only 2 turns.

Cubic polynomials have either 0 or 2 turns

Question 22

What is the x-intercept of

f(x) = x³ + 4

Answer 22

(0,4)

Question 23

How does the graph of

x³ - 3x² + 3x -1

compare to

x³ ?

Answer 23

x³ - 3x² + 3x -1 =

(x - 1)³

This is the graph of x³ moved 1 to the right.

Question 24

How many x-intercepts are there in:

f(x) = x⁴ - 9x²

Which direction do the ends go?

Answer 24

f(x) = x⁴ - 9x²

= x²(x² - 9)

= x²(x - 3)(x + 3)

3 roots: x = 0 and ±3

both ends point in the same direction (even exponent) and they go up (positive coefficient)

Question 25

What is the end behavior of

f(x) = 3x⁶ - 2x + 7

Answer 25

degree is even

leading exponent is positive

Both ends point up

Question 26

What is the end behavior of

f(x) = -4x⁵ + 3x⁴ + 2x²

Answer 26

degree = odd

lead coefficient = negative

left side points up

right side points down

Question 27

What is the end behavior of

f(x) = -10x⁸ + 20

Answer 27

degree = even

lead coefficient = negative

Both ends point down

Question 28

What is the end behavior of

f(x) = (x² - 1)²

Answer 28

degree = even

lead coefficient = positive

Both ends point up