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LiRa [457]
3 years ago
14

5. f(x) = x ^ 3 + p * x ^ 2 + qx + 6 (a) Find the value of each constants p and q, given that (x - 1) is a factor of f(x) and wh

en f(x) is divided by (x + 1) the remainder is 8. (b) Hence, solve the equation f(x) = 0
Mathematics
1 answer:
lara31 [8.8K]3 years ago
4 0

Answer:

\boxed{\textsf{ The value of p is \textbf{-2} and the value of q is \textbf{ -5}.}}

Step-by-step explanation:

A polynomial is given to us and we need to find the value of the constants in the given equation . The given polynomial to us is :-

\sf\implies f(x)= x^3+px^2+qx + 6

It's mentioned that (x - 1) is a factor of f(x) . This means on putting x = 1 in the given polynomial the value of the polynomial becomes 0 .

<u>Puttting</u><u> </u><u>x</u><u> </u><u>=</u><u> </u><u>1</u><u> </u><u>in</u><u> </u><u>f</u><u>(</u><u>x</u><u>)</u><u> </u><u>:</u><u>-</u>

\sf\implies f(x)= x^3+px^2+qx + 6 \\\\\sf\implies f(1) = 1^3 + p(1)^2+q(1) + 6 = 0 \\\\\sf\implies f(1)= 1+p + q + 6 = 0 \\\\\sf\implies \boxed{ \red{\sf p + q = -7 }}

\rule{200}2

Now secondly it's given that on dividing the given polynomial by (x + 1) , the remainder is 8 . This means that on putting x = -1 the value of the given polynomial becomes 8 .

<u>Putt</u><u>ting</u><u> </u><u>x</u><u> </u><u>=</u><u> </u><u>(</u><u>-</u><u>1</u><u>)</u><u> </u><u>:</u><u>-</u>

\sf\implies f(x)= x^3+px^2+qx + 6 \\\\\sf\implies f(-1) = (-1)^3 + p(-1)^2+q(-1)+6 = 8 \\\\\sf\implies f(-1)= -1+p -q + 6 = 8 \\\\\sf\implies p - q = 8 -6 +1 \\\\\sf\implies  \boxed{ \red{\sf p - q = 3 }}

\rule{200}2

<u>Adding</u><u> </u><u>the</u><u> </u><u>above</u><u> </u><u>two</u><u> </u><u>equations</u><u> </u><u>:</u><u>-</u>

\sf\implies p + q + p - q = -7 + 3 \\\\\sf\implies 2p = -4 \\\\\sf\implies p =\dfrac{-4}{2}\\\\\sf\implies \boxed{\pink{\frak{ p = (-2) }}}

\rule{200}2

<u>Put</u><u> </u><u>this</u><u> </u><u>value</u><u> </u><u>of</u><u> </u><u>p</u><u> </u><u>in</u><u> </u><u>equation</u><u> </u><u>(</u><u>i</u><u>)</u><u> </u><u>.</u>

\sf\implies p + q = -7 \\\\\sf\implies  -2 + q = -7 \\\\\sf\implies q = -7+2 \\\\\sf\implies \boxed{\pink{\frak{ q = (-5) }}}

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Question 9 of 15 (1 point)
mote1985 [20]

The number is -9

Step-by-step explanation:

Let x be the number then according to given statement

4x+10 = x-17

Subtracting 10 from both sides

4x+10-10 = x-17-10\\4x = x-27

Subtracting x from both sides

4x-x = x-x-27\\3x = -27

Dividing both sides by 3

\frac{3x}{3} = \frac{-27}{3}\\ x = -9

<u>Verification:</u>

4(-9)+10 = -9-17\\-36+10=-26\\-26=-26

The number is -9

Keywords: Linear equation, variables

Learn more about linear equation at:

  • brainly.com/question/4767370
  • brainly.com/question/4770453

#learnwithBrainly

7 0
2 years ago
the price of bread went from 1.50 per loaf to 2.50 per loaf in four years. find the rate of change of the price of bread
Talja [164]

Answer:

I think the answer is 1 I really hope this helps

Step-by-step explanation:


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3 years ago
I NEED AN ANSWER ASAP PLZZZ!!!
Helen [10]

Answer:

A.

Step-by-step explanation:

4 0
3 years ago
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Which equation is the inverse of 5y+4= (x+3)² + 2?
Neko [114]
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Kristin spent $120 on shirts. Dress shirts cost $40 and t-shirts cost $8. If she bought a total of 7 shirts, then how many of ea
FinnZ [79.3K]

Answer:

The correct answer is: She bought 2 dress shirts and 5 t-shirts

Step-by-step explanation:

Let's marked dress shirts with x and t-shirts with y.

Now we will make equations to solve this problem:

x + y = 7  =>  y = 7 - x

40 x + 8 y = 120

When we replace y = (7 - x) in the previous equation we get:

40 x + 8 (7 -x) = 120  40 x + 56 - 8 x = 120  => 32 x = 120 - 56 =>

32 x = 64  => x = 64/32 = 2 => x = 2

y = 7 - x = 7 - 2 = 5 =>  y = 5

x = 2 dress shirts

y = 5 t-shirts

God with you!!!

8 0
3 years ago
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