MHF 4U Lesson 2.0 Review of Factoring Ex. Factor each of the following completely. a) 2 4 3 35 28 42 p p p b) 10 3 2 x x c) 2 8 50 b d) 2 2 18 7 b ab a e) yz xz xy x 2 f) 15 7 2 2 x x g) 3 10 8 2 x x h) 2 2 49 28 4 b ab a i) 36 13 2 4 x x j) 2 2 ) ( 4 c b a k) 25 10 16 2 4 y y x WS 2.0
30
Embed
MHF 4U Lesson 2.0 Review of Factoring Ex. Factor each of ... · MHF 4U Lesson 2.2 The Factor and Remainder Theorems The Remainder Theorem The remainder theorem states: When a polynomial
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
MHF 4U Lesson 2.0 Review of Factoring
Ex. Factor each of the following completely.
a) 243 352842 ppp b) 1032 xx
c) 2850 b d) 22 187 baba
e) yzxzxyx 2 f) 1572 2 xx
g) 3108 2 xx h) 22 49284 baba
i) 3613 24 xx j) 22)(4 cba
k) 251016 24 yyx
WS 2.0
MHF 4U Lesson 2.1 Dividing Polynomials
Dividing polynomials can be done in more than one way. It is important to use the most efficient way
in order to solve problems in the simplest manner.
I. Dividing by Factoring.
Whenever it is possible to divide polynomials by factoring, it is the
simplest and most efficient way to solve the problem.
Failing to recognize this will cause you to waste time and effort
on a more inefficient method of solving the problem.
II. Dividing using Long Division
Ex. 1 a) Divide 12823 32 xxx by 1x and express your answer in quotient form.
b) State any restrictions on the variable.
c) Write a corresponding statement that can be used to check the division.
d) Verify your answer.
6
61122
x
xx
Ex. 2 a) Divide 1294 3 xx by 12 x and express your answer in quotient form.
b) Write a corresponding statement that can be used to check the division.
Ex. 3 The volume, V, in cubic centimeters, of a rectangular box is given by 8147)( 23 xxxxV .
Determine expressions for possible dimensions of the box if the height, h, in centimeters is given
by 2x .
Ex. Divide each of the following using synthetic division.
a) )2()32( 2 xxx b) )3()953( 2 xxx
c) )12()3648( 23 xxxx d) )1()1( 3 xx
Pg. 168 # 2, 3, 4, (5 – 10)doso, 11, 12
MHF 4U Lesson 2.2 The Factor and Remainder Theorems
The Remainder Theorem
The remainder theorem states: When a polynomial f(x), is divided by x – a, the remainder is equal to f(a).
Ex. 1 a) Given: f(x) = 2332 23 xxx evaluate each of the following.
(i) f(2) (ii) f(-3) (iii) f(1)
b) Divide by each of the following.
(i) 2x (ii) 3x (iii) 1x
The Factor Theorem
From the remainder theorem, we have seen that the remainder can be found by determining the value
of f(a). By extrapolating, we can determine that if the remainder is zero, the function is evenly divisible
by the divisor.
The factor theorem states: x – a is a factor of f(x) if and only if (iff) f(a) = 0.
Ex. 2 Determine whether or not 2x is a factor of 43)( 23 xxxxf .
Ex. 3 Factor completely.
a) 652 23 xxx b) 182773 234 xxxx
c) 2723 23 xxx
Ex. 4 When 22 23 nxmxx is divided by 1x the remainder is -12 and 2x is a factor. Determine the
values of m and n.
Ex. 5 If when 54 23 kxxx is divided by 2x the remainder is 7, what is the value of
k ?
Pg. 176 # (1 – 7)doso, 9, 10, 12, 14
MHF 4U Lesson 2.3 Sum and Difference of Cubes
A sum or difference of cubes is in the form 33 ba or 33 ba .
Ex. 1 Factor 33 ba using the factor theorem.
If we use the factor theorem on 33 ba , we can see that 33 ba = ))(( 22 bababa .