Question

Find the polynomial function

Answer 1

Answer:

y = 2x³ +x² -3x +6

Step-by-step explanation:

The polynomial correlation function of a graphing calculator can work with the table of function values to give you the equation. (See attached)

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If you want to solve this by hand, you can write the equations for the unknown coefficients in the polynomial function, then solve those. First of all, notice that you are given the y-intercept, (0, 6), so you know one of the coefficients already. That leaves a system of 3 equations in 3 unknowns that need to be solved.

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Writing the equations

... ax³ +bx² +cx +6 = y . . . . the generic linear equation in a, b, c that you're writing

For x = -1:

... -a +b -c +6 = 8

For x = 1

... a + b + c + 6 = 6

For x = 2

... 8a +4b +2c +6 = 20

In augmented matrix terms, this set of linear equations looks like ...

$\left[\begin{array}{ccc|c}-1&1&-1&2\\1&1&1&0\\8&4&2&14\end{array}\right]$

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Solving the equations

There are a number of ways to solve these. Again, a graphing calculator often has solution functions, such as reducing a matrix to row-echelon form, that will solve this in a keystroke or two.

If you add the first two equations, you get ...

... 2b = 2 ⇒ b = 1

Substituting that into the first equation allows you to simplify it to ...

... -a -c = 1

Dividing the third equation by 2 (and substituting for b) makes it ...

... 4a +c = 5

Adding these equations gives ...

... 3a = 6 ⇒ a = 2

Then ...

... c = -a -1 = -2 -1 = -3

And our polynomial function is ...

... y = 2x³ +x² -3x +6