P(x) is a polynomial. Here are a few values of $P(x)$ : $\newline$ $P(-6) = 3$ $\newline$ $P(-4) = -1$ $\newline$ $P(4) = 2$ $\newline$ $P(6) = -5$

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Find derivatives of polynomials

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Q. P(x) is a polynomial. Here are a few values of

P(x)

\newline

P(-6) = 3

\newline

P(-4) = -1

\newline

P(4) = 2

\newline

P(6) = -5

Write Given Values: Write down the given values. $P(-6) = 3$ , $P(-4) = -1$ , $P(4) = 2$ , $P(6) = -5$
Assume Polynomial Degree: Assume $P(x)$ is a polynomial of degree $3$ (cubic polynomial). $P(x) = ax^3 + bx^2 + cx + d$
Substitute $x$ Values: Substitute $x = -6$ into the polynomial equation. \[ 3 = a(-6)^3 + b(-6)^2 + c(-6) + d \] \[ 3 = -216a + 36b - 6c + d \]
Write System of Equations: Substitute $x = -4$ into the polynomial equation. $\newline$ $-1 = a(-4)^3 + b(-4)^2 + c(-4) + d$ $\newline$ $-1 = -64a + 16b - 4c + d$
Solve System of Equations: Substitute $x = 4$ into the polynomial equation. $2 = a(4)^3 + b(4)^2 + c(4) + d$ $2 = 64a + 16b + 4c + d$
Simplify Equation: Substitute $x = 6$ into the polynomial equation. $\newline$ $-5 = a(6)^3 + b(6)^2 + c(6) + d$ $\newline$ $-5 = 216a + 36b + 6c + d$
Solve for $b$ and $d$ : Write the system of equations. $\newline$ \[-216a + 36b - 6c + d = 3\] $\newline$ \[-64a + 16b - 4c + d = -1\] $\newline$ \[64a + 16b + 4c + d = 2\] $\newline$ \[216a + 36b + 6c + d = -5\]
Substitute $b$ to Find $d$ : Solve the system of equations. Add equations $1$ and $4$ : $-216a + 36b - 6c + d + 216a + 36b + 6c + d = 3 - 5$ $72b + 2d = -2$
Substitute $b$ and $d$ : Simplify the equation. $36b + d = -1$
Solve for $a$ and $c$ : Add equations $2$ and $3$ : $\newline$ $-64a + 16b - 4c + d + 64a + 16b + 4c + d = -1 + 2$ $\newline$ $32b + 2d = 1$
Solve System for $a$ and $c$ : Simplify the equation. $16b + d = 0.5$
Substitute $a$ to Find $c$ : Solve for $b$ and $d$ . From $36b + d = -1$ and $16b + d = 0.5$ : $36b + d - (16b + d) = -1 - 0.5$ $20b = -1.5$ $b = -0.075$
Substitute $a$ to Find $c$ : Solve for $b$ and $d$ . From $36b + d = -1$ and $16b + d = 0.5$ : $36b + d - (16b + d) = -1 - 0.5$ $20b = -1.5$ $b = -0.075$ Substitute $b$ back to find $d$ . $16(-0.075) + d = 0.5$ $-1.2 + d = 0.5$ $d = 1.7$
Substitute $a$ to Find $c$ : Solve for $b$ and $d$ . From $36b + d = -1$ and $16b + d = 0.5$ : $36b + d - (16b + d) = -1 - 0.5$ $20b = -1.5$ $b = -0.075$ Substitute $b$ back to find $d$ . $16(-0.075) + d = 0.5$ $-1.2 + d = 0.5$ $d = 1.7$ Substitute $b$ and $d$ into original equations to find $a$ and $c$ . $-216a + 36(-0.075) - 6c + 1.7 = 3$ $-216a - 2.7 - 6c + 1.7 = 3$ $-216a - 6c = 4$
Substitute $a$ to Find $c$ : Solve for $b$ and $d$ . From $36b + d = -1$ and $16b + d = 0.5$ : $36b + d - (16b + d) = -1 - 0.5$ $20b = -1.5$ $b = -0.075$ Substitute $b$ back to find $d$ . $16(-0.075) + d = 0.5$ $-1.2 + d = 0.5$ $d = 1.7$ Substitute $b$ and $d$ into original equations to find $a$ and $c$ . $-216a + 36(-0.075) - 6c + 1.7 = 3$ $-216a - 2.7 - 6c + 1.7 = 3$ $-216a - 6c = 4$ Solve for $a$ and $c$ . $-64a + 16(-0.075) - 4c + 1.7 = -1$ $-64a - 1.2 - 4c + 1.7 = -1$ $-64a - 4c = -1.5$
Substitute $a$ to Find $c$ : Solve for $b$ and $d$ . From $36b + d = -1$ and $16b + d = 0.5$ : $36b + d - (16b + d) = -1 - 0.5$ $20b = -1.5$ $b = -0.075$ Substitute $b$ back to find $d$ . $16(-0.075) + d = 0.5$ $-1.2 + d = 0.5$ $d = 1.7$ Substitute $b$ and $d$ into original equations to find $a$ and $c$ . $-216a + 36(-0.075) - 6c + 1.7 = 3$ $-216a - 2.7 - 6c + 1.7 = 3$ $-216a - 6c = 4$ Solve for $a$ and $c$ . $-64a + 16(-0.075) - 4c + 1.7 = -1$ $-64a - 1.2 - 4c + 1.7 = -1$ $-64a - 4c = -1.5$ Solve the system of equations for $a$ and $c$ . $-216a - 6c = 4$ $-64a - 4c = -1.5$
Substitute $a$ to Find $c$ : Solve for $b$ and $d$ . From $36b + d = -1$ and $16b + d = 0.5$ : $36b + d - (16b + d) = -1 - 0.5$ $20b = -1.5$ $b = -0.075$ Substitute $b$ back to find $d$ . $16(-0.075) + d = 0.5$ $-1.2 + d = 0.5$ $d = 1.7$ Substitute $b$ and $d$ into original equations to find $a$ and $c$ . $-216a + 36(-0.075) - 6c + 1.7 = 3$ $-216a - 2.7 - 6c + 1.7 = 3$ $-216a - 6c = 4$ Solve for $a$ and $c$ . $-64a + 16(-0.075) - 4c + 1.7 = -1$ $-64a - 1.2 - 4c + 1.7 = -1$ $-64a - 4c = -1.5$ Solve the system of equations for $a$ and $c$ . $-216a - 6c = 4$ $-64a - 4c = -1.5$ Simplify and solve. Multiply second equation by $1.5$ : $-96a - 6c = -2.25$ $-216a - 6c - (-96a - 6c) = 4 - (-2.25)$ $-120a = 6.25$ $a = -0.052$
Substitute $a$ to Find $c$ : Solve for $b$ and $d$ . From $36b + d = -1$ and $16b + d = 0.5$ : $36b + d - (16b + d) = -1 - 0.5$ $20b = -1.5$ $b = -0.075$ Substitute $b$ back to find $d$ . $16(-0.075) + d = 0.5$ $-1.2 + d = 0.5$ $d = 1.7$ Substitute $b$ and $d$ into original equations to find $a$ and $c$ . $-216a + 36(-0.075) - 6c + 1.7 = 3$ $-216a - 2.7 - 6c + 1.7 = 3$ $-216a - 6c = 4$ Solve for $a$ and $c$ . $-64a + 16(-0.075) - 4c + 1.7 = -1$ $-64a - 1.2 - 4c + 1.7 = -1$ $-64a - 4c = -1.5$ Solve the system of equations for $a$ and $c$ . $-216a - 6c = 4$ $-64a - 4c = -1.5$ Simplify and solve. Multiply second equation by $1.5$ : $-96a - 6c = -2.25$ $-216a - 6c - (-96a - 6c) = 4 - (-2.25)$ $-120a = 6.25$ $a = -0.052$ Substitute $a$ back to find $c$ . $-64(-0.052) - 4c = -1.5$ $3.328 - 4c = -1.5$ $-4c = -4.828$ $c = 1.207$

P(x) is a polynomial. Here are a few values of $P(x)$ : $\newline$ $P(-6) = 3$ $\newline$ $P(-4) = -1$ $\newline$ $P(4) = 2$ $\newline$ $P(6) = -5$

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P(x) is a polynomial. Here are a few values of P(x) P(x) P(x): \newlineP(−6)=3 P(-6) = 3 P(−6)=3 \newlineP(−4)=−1 P(-4) = -1 P(−4)=−1 \newlineP(4)=2 P(4) = 2 P(4)=2 \newlineP(6)=−5 P(6) = -5 P(6)=−5

P(x) is a polynomial. Here are a few values of $P(x)$ : $\newline$ $P(-6) = 3$ $\newline$ $P(-4) = -1$ $\newline$ $P(4) = 2$ $\newline$ $P(6) = -5$