Find the unique polynomial of degree 4 that takes on values $\mathit{p}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{}\mathit{p}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}\mathit{p}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathit{p}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathbf{,}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{p}\mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{=}\mathbf{0}$. Write your answer in the coefficient representation.

Here is just a representation of the generic distinct polynomial equation with degree "n":

$P\left(x\right)=P_0+P_{1}x+P_1{x}^2+...+P_n{x}^n$ …… (1)

This following is indeed the equation of a distinct polynomial with degree 4 derived using equation (1):

localid="1659770618378" $P\left(x\right)=P_0+P_{1}x+P_2{x}^2+P_3{x}^3+P_4{x}^4$ …… (2)

Substitute the value of x as “1” in equation (1)

$\begin{array}{rcl}P\left(1\right)& =& P_0+P_{1}1+P_2{1}^2+P_3{1}^3+P_4{1}^4\\ & =& P_0+P_1+P_2+P_3+P_4\end{array}$…… (3)

Substitute the value of P(1) as “2” in equation (3)

$\begin{array}{rcl}2& =& P_0+P_1+P_2+P_3+P_4\end{array}$ …… (4)

Substitute the value of x as “2” in equation (1)

role="math" localid="1659771467014" $\begin{array}{rcl}P\left(2\right)& =& P_0+P_{1}2+P_2{2}^2+P_3{2}^3+P_4{2}^4\\ & =& P_0+2P_1+4P_2+8P_3+16P_4\end{array}$ …… (5)

Substitute the value of P (2) as “1” in equation (5)

$\begin{array}{rcl}1& =& P_0+2P_1+4P_2+8P_3+16P_4\end{array}$ …… (6)

Substitute the value of x as “3” in equation (1)

role="math" localid="1659771489756" $\begin{array}{rcl}P\left(3\right)& =& P_0+P_{1}3+P_2{3}^2+P_3{3}^3+P_4{3}^4\\ & =& P_0+3P_1+9P_2+27P_3+81P_4\end{array}$ …… (7)

Substitute the value of P(3) as “0” in equation (7)

$\begin{array}{rcl}0& =& P_0+3P_1+9P_2+27P_3+81P_4\end{array}$ …… (8)

Substitute the value of as “4” in equation (1)

$\begin{array}{rcl}P\left(4\right)& =& P_0+P_{1}4+P_2{4}^2+P_3{4}^3+P_4{4}^4\\ & =& P_0+4P_1+16P_2+64P_3+256P_4\end{array}$ …… (9)

Substitute the value of P(4) as “4” in equation (9)

$\begin{array}{rcl}4& =& P_0+4P_1+16P_2+64P_3+256P_4\end{array}$ ……(10)

Substitute the value of X as “5” in equation (1)

role="math" localid="1659771676062" $\begin{array}{rcl}P\left(5\right)& =& P_0+P_{1}5+P_2{5}^2+P_3{5}^3+P_4{5}^4\\ & =& P_0+5P_1+25P_2+125P_3+625P_4\end{array}$ …… (11)

Substitute the value of P(5) as “0” in equation (11)

$\begin{array}{rcl}0& =& P_0+5P_1+25P_2+125P_3+625P_4\end{array}$ …… (12)

Find the coefficients by using cramer’s rule:

The formula for matrix representation is

$Ay=B$ …… (13)

As illustrated in equations (14), equation (15), and equation (16), the equations (4), (6), (8), (10), and (12) may be represented in matrix format as follows:

In matrix A, represent the variables as follows:

localid="1659772894874" $A=\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 4& 8& 16\\ 1& 3& 9& 27& 81\\ 1& 4& 16& 64& 256\\ 1& 5& 25& 125& 625\end{array}\right]\dots \dots \left(14\right)$

Represent the coefficients localid="1659772510651" $P_0,P_1,P_2,P_3,P_4$in matrix y as follows:

$Y=\left[\begin{array}{c}{P}_0\\ P_1\\ P_2\\ P_3\\ P_4\end{array}\right]\dots \dots \left(15\right)$

In matrix B, express those values of both the equation as follows:

$B=\left[\begin{array}{c}2\\ 1\\ 0\\ 4\\ 0\end{array}\right]\dots \dots \left(16\right)$

Substitute equation (14), equation (15) and equation (16) in equation (13).

$\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 4& 8& 16\\ 1& 3& 9& 27& 81\\ 1& 4& 16& 64& 256\\ 1& 5& 25& 125& 625\end{array}\right]\times \left[\begin{array}{c}{P}_0\\ P_1\\ P_2\\ P_3\\ P_4\end{array}\right]=\left[\begin{array}{c}2\\ 1\\ 0\\ 4\\ 0\end{array}\right]\dots \dots \left(17\right)$

Determine that determinant of Either a, and divide each determinant of the coefficient by the determinant of A to find the values of the coefficients $P_0,P_1,P_2,P_3,P_4$as follows:

The Gaussian elimination approach may be used to find A's determinant, which is 288.

$∆A=288.....\left(18\right)$

Next, find the coefficient of P₀.

To find the value of P₀ , replace the first column of A by B. Thus,

$P_0=\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 4& 8& 16\\ 1& 3& 9& 27& 81\\ 1& 4& 16& 64& 256\\ 1& 5& 25& 125& 625\end{array}\right]$

The Gaussian elimination approach may be used to find P₀'s determinant, which is -5760.

$∆P_0=-5760..........\left(19\right)$

Find the coefficient of P₀ as,

role="math" localid="1659773416538" $Coeff.of{P}_0=\frac{∆P_0}{∆A}.........\left(20\right)$

Use equation (18) and equation (19) in equation (20).

role="math" localid="1659773484679" $$\begin{gathered} Coeff.of{P}_0=\frac{-5760}{288} \\ Coeff.of{P}_0=-20...........\left(21\right) \end{gathered}$$

Thus, the coefficient of P₀ is -20.

Next, find the coefficient of P₁ .

To find the value of P₁, replace the second column of A by B. Thus,

role="math" localid="1659773613814" $P_1=\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 4& 8& 16\\ 1& 3& 9& 27& 81\\ 1& 4& 16& 64& 256\\ 1& 5& 25& 125& 625\end{array}\right]$

The Gaussian elimination approach may be used to find P₁'s determinant, which is 13,152.

$∆P_1=13,152.........\left(22\right)$

Find the coefficient of P₁ as,

role="math" localid="1659773738779" $Coeff.of{P}_1=\frac{∆P_1}{∆A}.........\left(23\right)$

Use equation (18) and equation (22) in equation (23).

role="math" localid="1659773783405" $$\begin{gathered} Coeff.of{P}_1=\frac{13,152}{288} \\ Coeff.of{P}_1=45.66...........\left(24\right) \end{gathered}$$

Thus, the coefficient of P₁ is 45.66.

Next, find the coefficient of P₂.

To find the value of P₂, replace the third column of A by B. Thus,

role="math" localid="1659774249002" $P_2=\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 4& 8& 16\\ 1& 3& 9& 27& 81\\ 1& 4& 16& 64& 256\\ 1& 5& 25& 125& 625\end{array}\right]$

The Gaussian elimination approach may be used to find P₂'s determinant, which is -9000.

$∆P_2=-9000.......\left(25\right)$

Find the coefficient of P₂ as,

role="math" localid="1659774340068" $Coeff.of{P}_2=\frac{∆P_2}{∆A}.........\left(26\right)$

Use equation (18) and equation (25) in equation (26).

role="math" localid="1659774381998" $$\begin{gathered} Coeff.of{P}_2=\frac{-9000}{288} \\ Coeff.of{P}_2=-31.25.........\left(27\right) \end{gathered}$$

Thus, the coefficient of P₂ is -31.25.

Next, find the coefficient of P₃.

To find the value of P₃, replace the fourth column of A by B. Thus,

$P_3=\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 4& 8& 16\\ 1& 3& 9& 27& 81\\ 1& 4& 16& 64& 256\\ 1& 5& 25& 125& 625\end{array}\right]$

The Gaussian elimination approach may be used to find P₃'s determinant, which is 2400.

$∆P_3=2500.......\left(28\right)$

Find the coefficient of P₃ as,

role="math" localid="1659774519843" $Coeff.of{P}_3=\frac{∆P_3}{∆A}.........\left(29\right)$

Use equation (18) and equation (28) in equation (29).

role="math" localid="1659774579633" $$\begin{gathered} Coeff.of{P}_3=\frac{2400}{288} \\ Coeff.of{P}_3=8.33.........\left(30\right) \end{gathered}$$

Thus, the coefficient of P₃ is 8.33.

Similarly, the determinant of the matrix P₄ is represented as follows:

$∆P_4=-216.....\left(31\right)$

Substitute coefficient as P₄ in equation (18).

$Coeff.of{P}_4=\frac{∆P_4}{∆A}.........\left(32\right)$

Use equation (18) and equation (31) in equation (32).

$$\begin{gathered} Coeff.of{P}_3=\frac{-216}{288} \\ Coeff.of{P}_2=-0.75.........\left(33\right) \end{gathered}$$

Thus, the coefficient of P₄ is -0.75.

Therefore, the coefficient representation of unique polynomial equation $P_0,P_1,P_2,P_3,and{P}_{4}are-20,45.66,-31.25,8.33$and, -0.75.