Given f(x) =x^2 -x and g(x)=3x-4 what is g(f(-3))

Please help with these partial fractions!!!

VARVARA [1.3K]

a. Factorize the denominator:

$\dfrac{x+14}{x^2-2x-8}=\dfrac{x+14}{(x-4)(x+2)}$

Then we're looking for $a,b$ such that

$\dfrac{x+14}{x^2-2x-8}=\dfrac a{x-4}+\dfrac b{x+2}$

$\implies x+14=a(x+2)+b(x-4)$

If $x=4$ , then $18=6a\implies a=3$ ; if $x=-2$ , then $12=-6b\implies b=-2$ . So we have

$\dfrac{x+14}{x^2-2x-8}=\dfrac3{x-4}-\dfrac2{x+2}$

as required.

b. Same setup as in (a):

$\dfrac{-3x^2+5x+6}{x^3+x^2}=\dfrac{-3x^2+5x+6}{x^2(x+1)}$

We want to find $a,b,c$ such that

$\dfrac{-3x^2+5x+6}{x^2(x+1)}=\dfrac ax+\dfrac b{x^2}+\dfrac c{x+1}$

Quick aside: for the second term, since the denominator has degree 2, we should be looking for another constant $b'$ such that the numerator of the second term is $b'x+b$ . We always want the polynomial in the numerator to have degree 1 less than the degree of the denominator. But we would end up determining $b'=0$ anyway.

$\implies-3x^2+5x+6=ax(x+1)+b(x+1)+cx^2$

If $x=0$ , then $b=6$ ; if $x=-1$ , then $c=-2$ . Expanding everything on the right then gives

$-3x^2+5x+6=ax^2+ax+bx+b+cx^2=(a-2)x^2+(a+6)x+6$

which tells us $a-2=-3$ and $a+6=5$ ; in both cases, we get $a=-1$ . Then

$\dfrac{-3x^2+5x+6}{x^2(x+1)}=-\dfrac1x+\dfrac6{x^2}-\dfrac2{x+1}$

as required.

5 0

4 years ago

Translate the sentence into an equation.

SpyIntel [72]

7+(w+3)=5
This is the answer to your question

6 0

3 years ago

Kenny has 8 red shirts, 12 blue shirts, and 10 white shirts in his closet. What is the probability that he would get a white shi

antiseptic1488 [7]

Answer:

1/3 or 33.33 %

Step-by-step explanation:

this is because there is 30 shirts and all and 10 white. 10/30 is 1/3, or 33.33%

8 0

3 years ago

Read 2 more answers

Evaluate when x = -3 and y= -1:<br><br><br> x-xy=

LenKa [72]

-3-(-3)(-1)= 0, so 0 is the solution

5 0

3 years ago

Read 2 more answers

If the average, mode, and range of 5 different numbers are all 8, find the sum of all possible values of the second largest numb

ANTONII [103]

I believe you meant:

If the average, mode, and range of 5 different numbers are all 8, find the sum of all possible values and the second largest number

Answer:

Sum of all possible values = 40

Second largest number= 8

Explanation:

Since our mean, mode, and range is 8, our sum is 40 and our second largest number is 8.

The sum of all the numbers is 40 since our mean is 8 and there are 5 numbers(8×5)

To find possible values, we already know the mode is 8 and so we know 8 must appear more than once. If we have 8 twice then we are left with 24(sum is 40). We have 3 numbers left and the range is 8 so we must have a maximum number and minimum number that has a difference of 8 and a third number, all adding up to 24(sum left). Within 24, only 12-4+8 satisfies the condition, hence our numbers are: 4,8,8,8,12.

3 0

3 years ago