### Instruction: Find the cubic polynomial $p(x)$ such that $p(1) = -7,$ $p(2) = -9,$ $p(3) = -15,$ and $p(4) = -31.$.### Response: I need to find a cubic polynomial of the form $p(x) = ax^3 + bx^2 + cx + d$ that satisfies the given values of $p(1),$ $p(2),$ $p(3),$ and $p(4).$
I can plug in each value of $x$ and $p(x)$ into the equation and get a system of four linear equations in four variables: $a, b, c,$ and $d.$
The system is: \begin{align*} a + b + c + d &= -7 \\ 8a + 4b + 2c + d &= -9 \\ 27a + 9b + 3c + d &= -15 \\ 64a + 16b + 4c + d &= -31 \end{align*}
I can use any method to solve this system, such as elimination or substitution.
I will try to eliminate one variable at a time by subtracting pairs of equations.
Subtracting the first equation from the second, I get: \begin{align*} 7a + 3b + c &= -2 \end{align*}
Subtracting the second equation from the third, I get: \begin{align*} 19a + 5b + c &= -6 \end{align*}
Subtracting the third equation from the fourth, I get: \begin{align*} 37a + 7b + c &= -16 \end{align*}
Now I have a system of three equations in three variables: $a, b,$ and $c.$
I can repeat the elimination process by subtracting pairs of equations.
Subtracting the first equation from the second, I get: \begin{align*} 12a + 2b &= -4 \end{align*}
Subtracting the second equation from the third, I get: \begin{align*} 18a + 2b &= -10 \end{align*}
Now I have a system of two equations in two variables: $a$ and $b.$
I can eliminate $b$ by subtracting the first equation from the second, and get: \begin{align*} 6a &= -6 \end{align*}
Dividing both sides by $6,$ I get: \begin{align*} a &= -1 \end{align*}
Now that I have $a,$ I can plug it into either equation to find $b.$
Using the first equation, I get: \begin{align*} 12(-1) + 2b &= -4 \\ -12 + 2b &= -4 \\ 2b &= 8 \\ b &= 4 \end{align*}
Now that I have $a$ and $b,$ I can plug them into either equation to find $c.$
Using the first equation, I get: \begin{align*} 7(-1) + 3(4) + c &= -2 \\ -7 + 12 + c &= -2 \\ 5 + c &= -2 \\ c &= -7 \end{align*}
Now that I have $a, b,$ and $c,$ I can plug them into any equation to find $d.$
Using the first equation, I get: \begin{align*} -1 + 4 - 7 + d &= -7 \\ -4 + d &= -7 \\ d &= -3 \end{align*}
Now I have all the coefficients of the cubic polynomial: $a = -1, b = 4, c = -7,$ and $d = -3.$
Therefore, the polynomial is: \begin{align*} p(x) &= -x^3 + 4x^2 - 7x - 3 \end{align*}