Pls help fast!!! giving 20 points!!!

A fourth-order polynomial passes through the five points for which the independent and dependent variables, x and y, respectivel

Sladkaya [172]

Answer:

The fourth-order polynomial that passes through the points is $P(x)=2x^4+x^3+3x^2+2x+5$

Step-by-step explanation:

A fourth-order polynomial has this general form:

$P(x)=ax^4+bx^3+cx^2+dx+e$

We need to replace the given points one by one, which leads to a system of 5 equations and 5 variables (namely a,b,c,d,e)

$37=a(-2)^4+b(-2)^3+c(-2)^2+d(-2)+e\\7=a(-1)^4+b(-1)^3+c(-1)^2+d(-1)+e\\5=a(0)^4+b(0)^3+c(0)^2+d(0)+e\\13=a(1)^4+b(1)^3+c(1)^2+d(1)+e\\61=a(2)^4+b(2)^3+c(2)^2+d(2)+e$

We can rewrite this system as follows:

$16a-8b+4c-2d+e=37\\a-b+c-d+e=7\\e=5\\a+b+c+d+e=13\\16a+8b+4c+2d+e=61$

We can use the Gaussian Elimination to solve this system of equations. To use the Gaussian Elimination we need to express the system of linear equations in matrix form (the matrix equation Ax = b).

The coefficient matrix (A) for the above system is

$\left[\begin{array}{ccccc}16&-8&4&-2&1\\1&-1&1&-1&1\\0&0&0&0&1\\1&1&1&1&1\\16&8&4&2&1\end{array}\right]$

the variable matrix (x) is

$\left[\begin{array}{c}a&b&c&d&e\end{array}\right]$

and the constant matrix (b) is

$\left[\begin{array}{c}37&7&5&13&61\end{array}\right]$

The augmented matrix for this system, it is obtained by appending the columns of the coefficient matrix and the constant matrix.

$\left[\begin{array}{ccccc|c}16&-8&4&-2&1&37\\1&-1&1&-1&1&7\\0&0&0&0&1&5\\1&1&1&1&1&13\\16&8&4&2&1&61\end{array}\right]$

The augmented matrix can be transformed by a sequence of elementary row operations to the reduced row echelon form.

$\left[\begin{array}{ccccc|c}1&0&0&0&0&2\\0&1&0&0&0&1\\0&0&1&0&0&3\\0&0&0&1&0&2\\0&0&0&0&1&5\end{array}\right]$

Therefore the solutions are:

$\left\begin{array}{c}a=2&b=1&c=3&d=2&e=5\end{array}\right$

And the polynomial P(x) is:

$P(x)=2x^4+x^3+3x^2+2x+5$

We can check our solution plotting the polynomial and checking that it passes through the points.

4 0

3 years ago

At New City High School, 11 out of 20 high school seniors have a part-time job. This ratio is the same at Oldville High School,

Law Incorporation [45]

Answer:

264 out of 480

Step-by-step explanation:

6 0

3 years ago

Mariah is debating whether to use a straightedge and compass or a computer drawing program to complete a construction of a regul

marshall27 [118]

I think the best way is to use a computer drawing program in constructing a regular hexagon inscribed in a circle. Its because a computer program is an accurate, precise and the most important it is more efficient than using the straight edge and compass

7 0

3 years ago

Read 2 more answers

Graph the number on a number line sqrt 11

LekaFEV [45]

Sqrt 11 is approximately 3.31662479036

So make a line from about -5 to 5 and just estimate where that could be, between the numbers 3 and 4, make sure you mark that that is sqrt 11

6 0

3 years ago

Write an ordered pair for each point graphed below.

Viefleur [7K]

Answer:

$\boxed{(-1,4)}$