Use synthetic substitution to find \(f(3)\) for each polynomial function
\(\textbf{1)}\) Find \( f(3) \) for \( f(x)=4x^4-3x^2+8x-2 \)
The answer is \( f(3)=319 \)
\(\textbf{2)}\) Find \( f(3) \) for \( f(x)=x^5-3x^4-4x^3-2x^2+4x+2 \)
The answer is \( f(3)=-112 \)
\(\textbf{3)}\) Find \( f(3) \) for \( f(x)=4x^4+2x^2-x-3\)
\(\text{The answer is }f(3)=336\)
\[ \begin{array}{r|rrrrr} 3 & 4 & 0 & 2 & -1 & -3 \\ & & 12 & 36 & 114 & 339 \\ \hline & 4 & 12 & 38 & 113 & 336 \\ \end{array} \] We use synthetic substitution with \(x = 3\) and the coefficients of \(4x^4 + 0x^3 + 2x^2 – x – 3\). The result is \( f(3) = 336 \).
\(\textbf{4)}\) Find \( f(1) \) for \( f(x)=2x^4 – 3x^3 + x^2 + 2x – 5\)
\(\text{The answer is }f(1)=-3\)
\[ \begin{array}{r|rrrrr} 1 & 2 & -3 & 1 & 2 & -5 \\ & & 2 & -1 & 0 & 2 \\ \hline & 2 & -1 & 0 & 2 & -3 \\ \end{array} \] We use synthetic substitution with \(x = 1\) and the coefficients of \(2x^4 – 3x^3 + x^2 + 2x – 5\). The result is \( f(1) = -3 \).
\(\textbf{5)}\) Find \( f(-2) \) for \( f(x)=x^4 + 3x^3 – 2x + 4\)
\(\text{The answer is }f(-2)=0\)
\[ \begin{array}{r|rrrrr} -2 & 1 & 3 & 0 & -2 & 4 \\ & & -2 & -2 & 4 & -4 \\ \hline & 1 & 1 & -2 & 2 & 0 \\ \end{array} \] We use synthetic substitution with \(x = -2\) and the coefficients of \(x^4 + 3x^3 + 0x^2 – 2x + 4\). The result is \( f(-2) = 0 \).
\(\textbf{6)}\) Find \( f(2) \) for \( f(x)=3x^4 – 5x^2 + 6x – 1\)
\(\text{The answer is }f(2)=39\)
\[ \begin{array}{r|rrrrr} 2 & 3 & 0 & -5 & 6 & -1 \\ & & 6 & 12 & 14 & 40 \\ \hline & 3 & 6 & 7 & 20 & 39 \\ \end{array} \] We use synthetic substitution with \(x = 2\) and the coefficients of \(3x^4 + 0x^3 – 5x^2 + 6x – 1\). The result is \( f(2) = 39 \).
\(\textbf{7)}\) Find \( f(-3) \) for \( f(x)=x^4 + 2x^3 – 5x – 8\)
\(\text{The answer is }f(-3)=34\)
\[ \begin{array}{r|rrrrr} -3 & 1 & 2 & 0 & -5 & -8 \\ & & -3 & 3 & -9 & 42 \\ \hline & 1 & -1 & 3 & -14 & 34 \\ \end{array} \] We use synthetic substitution with \(x = -3\) and the coefficients of \(x^4 + 2x^3 + 0x^2 – 5x – 8\). The result is \( f(-3) = 34 \).
\(\textbf{8)}\) Find \( f(4) \) for \( f(x)=5x^4 – 2x^3 + 3x – 9\)
\(\text{The answer is }f(4)=1155\)
\[ \begin{array}{r|rrrrr} 4 & 5 & -2 & 0 & 3 & -9 \\ & & 20 & 72 & 288 & 1164 \\ \hline & 5 & 18 & 72 & 291 & 1155 \\ \end{array} \] We use synthetic substitution with \(x = 4\) and the coefficients of \(5x^4 – 2x^3 + 0x^2 + 3x – 9\). The result is \( f(4) = 1155 \).
\(\textbf{9)}\) Find \( f(-1) \) for \( f(x)=2x^4 – 3x^3 + 4x^2 – 2x + 1\)
\(\text{The answer is }f(-1)=12\)
\[ \begin{array}{r|rrrrr} -1 & 2 & -3 & 4 & -2 & 1 \\ & & -2 & 5 & -9 & 11 \\ \hline & 2 & -5 & 9 & -11 & 12 \\ \end{array} \] We use synthetic substitution with \(x = -1\) and the coefficients of \(2x^4 – 3x^3 + 4x^2 – 2x + 1\). The result is \( f(-1) = 12 \).
\(\textbf{10)}\) Find \( f(5) \) for \( f(x)=3x^4 – 2x^2 + x – 7\)
\(\text{The answer is }f(5)=1823\)
\[ \begin{array}{r|rrrrr} 5 & 3 & 0 & -2 & 1 & -7 \\ & & 15 & 75 & 365 & 1830 \\ \hline & 3 & 15 & 73 & 366 & 1823 \\ \end{array} \] We use synthetic substitution with \(x = 5\) and the coefficients of \(3x^4 + 0x^3 – 2x^2 + x – 7\). The result is \( f(5) = 1823 \).
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