Properties of Polynomial Functions - Weeblymrjtaylor.weebly.com/uploads/3/9/7/2/39723380/mhf_4ui... · 2018. 10. 17. · 4. When x3 + 3x2 + kx + 10 is divided by x – 5, the remainder

MHF 4UI Unit 2 Day 1

Properties of Polynomial Functions

Terminology

The degree of a polynomial is ____________________________________________

Example Degree Classification

y 5x 3

2y 4x 3x 1

3y 4x 3

4 3y 5x 3x 2x

5y 7x 4x

The leading coefficient is _______________________________________________

___________________________________________________________________

A function is increasing if _______________________________________________

___________________________________________________________________

A function is decreasing if _______________________________________________

___________________________________________________________________

A turning point occurs when ______________________________________________

___________________________________________________________________

The “end behaviours” of a function means:

_________________________________________________________

As x approaches negative infinity,

___________________________________________________________________

Using mathematical notation: __________________________________

__________________________________

As x approaches positive infinity,

___________________________________________________________________

Using mathematical notation: ___________________________________

___________________________________

The zero of a function is ________________________________________________

___________________________________________________________________


Graphing Polynomial Functions in Factored Form

1. Sketch each function and label the x-intercepts.

a) y x 2 x 5

zeroes: _______________

degree: ________

sign of leading coefficient: __________

end behaviours: as x , f x _________

as x , f x _________

b) 2

y 2 x 1

zeroes: _______________

degree: ________



as x , f x _________

A double root ______________________________________________________

__________________________________________________________________

c) y x 2 x 1 x 2

zeroes: _______________

degree: ________



as x , f x _________


d) 2

y 4 x 3 x 2

zeroes: _______________

degree: ________



as x , f x _________

e) y 2 x 5 x 2 x 3 x 1

zeroes: _______________

degree: ________



as x , f x _________

f) 2

y 3 x 1 x 3 x 2

zeroes: _______________

degree: ________



as x , f x _________


g) 3 2

y 7 x 3 x 2

zeroes: _______________

degree: ________



as x , f x _________

A triple root _______________________________________________________

__________________________________________________________________

2. Sketch the graph of a polynomial function that satisfies each set of conditions.

a) degree 3, two zeroes, two turning points, positive leading coefficient

b) degree 3, one zero, no turning points, negative leading coefficient

c) degree 4, two zeroes, three turning points, negative leading coefficient


Cubic Functions: y = ax3 + bx2 +cx + d

End behaviours: if a > 0, as x - , f(x) -

as x + , f(x) +

if a < 0, as x - , f(x) +

as x + , f(x) -

Possible roots:

three distinct roots (three roots, three zeros)

two equal real roots and one distinct real root (three roots, two zeros)

one distinct real root and two complex roots (three roots, one zero)

three equal real roots (three roots, one zero)


x

y

Family of Functions

The graphs of the polynomial functions of the form y = k (x – x1)(x – x2) … (x – xn)

have the same roots (not x-intercepts) but different y-intercepts.

A family of functions is a set of functions that can be written in the same form.

1. a) Sketch f(x) = (x + 2)(x – 1)(x – 3).

b) Sketch on the same grid: g(x) = 2 (x + 2)(x – 1)(x – 3)

h(x) = - (x + 2)(x – 1)(x – 3)

Note: From above, we must say “the same roots” and not “the same x-intercepts”

since imaginary roots aren’t x-intercepts.

Example: y = 3(x – 2)(x + 3)2

is not in the same family as

y = 3(x – 2)(x + 3)2(x – i)(x + i)

they have different roots yet the same x-intercepts!


2. Write the equation, in standard form, of the family of polynomial functions

given each set of roots.

a) 4, -1, 3

2

b) -2, 34 , 3-4

3. Determine the equation of the cubic function, in standard form, with the roots

-3, -1 and 1 which passes through the point (2, -45).


4. Determine the equation of the quartic function, in standard form, with the

roots 2, -2, 3

i25 ,

3

i2-5 with y-intercept -216.


Division of Polynomials

1. Divide the following using long division:

a) 72 ÷ 3 b) 83 ÷6

c) (x2 – 7x – 18) ÷ (x – 9)

d) (3x4 – 2x3 + 4x2 + 8x + 9) ÷ (x2 – x + 1)

e) (8x3 + 4x2 – 14) ÷ (2x – 3)


f) (4x – 7 + 6x2 + 4x4 – 8x3) ÷ (2x2 – 1)

Summary:

Write the division statement in the following form:

P(x) = D(x) · Q(x) + R(x) or

D(x)

R(x)Q(x)

D(x)

P(x)


Quartic Functions: edxcxbxaxy 234

End behaviours:

If a > 0

If a < 0

Possible Roots Four Distinct Real Roots

Two Equal Real Roots and Two

Distinct Real Roots

Two Pairs of Equal Real Roots

Two Equal Real Roots and Two

Complex Roots

Four Equal Real Roots

Four Distinct Complex Roots

One Distinct Real Root and

Three Equal Real Roots

Two Equal Pairs of Complex

Roots

Two Distinct Real Roots and

Two Complex Roots


Quintic Functions: fexdxcxbxaxy 2345

End behaviours:

If a > 0

If a < 0

Possible Roots Five Distinct Real Roots

Two Pairs of Equal Real Roots and

One Distinct Real Root

Three Distinct Real Roots and Two

Complex Roots

Two Equal Real Roots, Two Complex

Roots and 1 Distinct Real Root

Four Complex Roots and One

Distinct Real Root

Two Equal Real Roots and Three

Distinct Real Roots

Four Equal Real Roots and One

Distinct Real Root

Five Equal Real Roots

Three Equal Real Roots and Two

Complex Roots


Equal Real Roots


Distinct Real Roots

One Distinct Real Root and Two

Equal Pairs of Complex Roots


The Remainder Theorem

1. For f(x) = 3x3 – 2x2 – 4x + 3, divide by

a) x + 2 b) x – 1

c) 3x + 1

3x3 – 2x2 – 4x + 3 =


2. Evaluate for f(x) = 3x3 – 2x2 – 4x + 3

a) f(-2) b) f(1) c) f(3

1 )

Note:

The Remainder Theorem:

If f(x) is divided by x – p, the remainder is f(p).

If f(x) is divided by qx-p, the remainder is f(q

p).

Why does this work?

3. Determine the remainder without performing long division.

a) (2x3 + 3x2 – 7x – 3) ÷ (x + 1) b) (3m3 + 2m2 +5m – 4) ÷ (2m - 3)


4. When x3 + 3x2 + kx + 10 is divided by x – 5, the remainder is 15. Determine the

value of k.

5. When the polynomial P(x) = 3x3 + cx2 + dx – 7 is divided by x – 2, the remainder

is -3. When P(x) is divided by x + 1, the remainder is -18. Determine the values

of c and d.

6. Determine the value of k given that x + 4 is a factor of 3x3 + 11x2 – 6x + k.


The Factor Theorem

The Factor Theorem: x – a is a factor of f(x) if and only if f(a) = 0.

Note: We can also read this as: if f(a) = 0, then x – a is a factor!

1. Factor fully.

a) x3 – x2 – 14x + 24

How do we know what numbers to try to determine a factor?


b) x3 – 2x2 – 5x + 6

c) x3 – 5x2 + 2x + 8


d) x3 + 2x2 – 7x – 8

e) x4 + 2x3 – 10x2 – 11x + 30


f) x4 - 2x3 – 7x2 + 8x + 12


The Factor Theorem - Extended

1. Factor fully.

21x2 + x - 10

This means that for f(x) = 21x2 + x – 10 then f( ) = 0 and f( ) = 0.

We would not easily guess these numbers.

How do we start guessing values for f(x) = ax2 + bx + c when a ≠ 1?

Let’s look at the roots to find some patterns.

7

5and

3

2 ; where to the 2 and -5 come from?

where to the 3 and 7 come from?

So, if the leading coefficient is an integer other than 1, there will be at least one factor

of the form qx – p.

But, by the Factor theorem, 0q

pf

, where p is a factor of the constant term

and q is a factor of the leading coefficient


2. Factor fully.

a) 3x3 – 19x2 + 27x – 7

b) 4w3 – 7w – 3

Documents

Properties of Polynomial Functions - Weeblymrjtaylor.weebly.com/uploads/3/9/7/2/39723380/mhf_4ui... · 2018. 10. 17. · 4. When x3 + 3x2 + kx + 10 is divided by x – 5, the remainder