1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
GuDViN [60]
3 years ago
8

If (x^2 - 1) is a factor of ax^4 + bx^3 + cx^2 + dx + e, show that a + c + e = b + d = 0

Mathematics
1 answer:
alisha [4.7K]3 years ago
8 0
x^2-1=x^2-1^2=(x-1)(x+1)

If x²-1 is a factor of the polynomial, both x-1 and x+1 are factors of it.

According to the remainder theorem, if a binomial x-a is a factor of a polynomial p(x), then p(a)=0.

If x-1 and x+1 are factors of the polynomial p(x)=ax⁴+bx³+cx²+dx+e, then p(1)=0 and p(-1)=0.

p(1)=a \times 1^4+b \times 1^3 + c \times 1^2+ d \times 1+e \\
p(-1)=a \times (-1)^4 + b \times (-1)^3 + c \times (-1)^2 + d \times (-1)+e \\ \\
p(1)=a+b+c+d+e \\
p(-1)=a-b+c-d+e \\ \\
p(1)=0 \\
p(-1)=0 \\ \\ \hbox{add both equations:} \\
a+b+c+d+e=0 \\
\underline{a-b+c-d+e=0} \\
2a+2c+2e=0 \\
2(a+c+e)=0 \\
a+c+e=0 \\ \\
\hbox{substitute 0 for a+c+e in the first equation:} \\
a+b+c+d+e=0 \\
(a+c+e)+b+d=0 \\
0+b+d=0 \\
b+d=0 \\ \\
\boxed{a+c+e=b+d=0} \\
\hbox{proved } \checkmark
You might be interested in
I will give brainlest thanks
Dmitrij [34]
The answer is choice A- 68
7 0
3 years ago
Read 2 more answers
What is 55 divided by 1 1/2
Karolina [17]

Answer:

55 divided by 11/2 is 10

5 0
3 years ago
Read 2 more answers
Can u help??????????????????????????????????????//
maw [93]

Answer:

Okay!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

The answer is 14

Step-by-step explanation:

Folow order of operations and when doing multipactiona or subtraction or division and addition, make sure to go from left to right

5 0
3 years ago
Read 2 more answers
Simplify <br>it was x there so u not confused​
Mariulka [41]

Step-by-step explanation:

here's the solution: -

=》

\frac{x {}^{6} }{ {x}^{4} }  \times  {x}^{2}

=》

\frac{x \times x \times x \times x \times x \times x}{x  \times x \times x\times \: x }  \times x \times x

=》

x \times x \times x \times x

=》

{x}^{4}

5 0
3 years ago
What are the next two numbers? 0.3 , 0.7 , 1.1 , 1.5 ,
Dmitry [639]
There is an increment of 0.4 in each term. Next will be
[1.5+0.4], [1.5+0.4+0.4]
1.9, 2.3
3 0
3 years ago
Read 2 more answers
Other questions:
  • At Ken's boutique, 33 customers wore a size small, and 51 customers wore a different size.
    5·2 answers
  • Eric and Mark were at a ball game and visited the concession stand. They each got a hamburger, and Mark got lemonade and some co
    14·1 answer
  • Stairs that measure 22 feet long are used to let passengers off an airplane. The distance from the bottom of the stairs to the p
    7·2 answers
  • In STU, SU =18 and TU =13. Find m
    13·1 answer
  • Find the slope of the line that passes through (9,5) and (5, 4).
    15·1 answer
  • Can u solve two questions. .. Mark brilliant student​
    12·1 answer
  • Describe how Malcolm X inspired the Black Power Movement.
    14·1 answer
  • Four friends attend a school play and pay $6.75 per ticket. Each also buys a Healthy Snack Bag sold by the Theater Club. If the
    5·2 answers
  • Looking for more friends(whats 2+2)<br> plz dont report
    5·2 answers
  • eric is flying an airplane at an altitude of 2200 feet. he sees his house on the ground at a 45⁰ angle of depression. what is Er
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!