Synthetic substitution is a shortcut for evaluating polynomial functions. Instead of plugging a value into every power of \(x\), we use the coefficients and repeat a multiply-add pattern. This is especially useful when polynomials have many terms or missing powers. It also connects nicely to synthetic division and the remainder theorem.
Use synthetic substitution to find the value of each polynomial function
\(\textbf{1)}\) Find \( f(3) \) for \( f(x)=4x^4-3x^2+8x-2 \)
The answer is \( f(3)=319 \)
\(\begin{array}{r|rrrrr} 3 & 4 & 0 & -3 & 8 & -2 \\ & & 12 & 36 & 99 & 321 \\ \hline & 4 & 12 & 33 & 107 & 319 \end{array}\)\(\text{Use synthetic substitution with }x=3\text{ and coefficients }4,0,-3,8,-2.\)\(\text{The final number is the value of the function.}\)\(f(3)=319\)
\(\textbf{2)}\) Find \( f(3) \) for \( f(x)=x^5-3x^4-4x^3-2x^2+4x+2 \)
The answer is \( f(3)=-112 \)
\(\begin{array}{r|rrrrrr} 3 & 1 & -3 & -4 & -2 & 4 & 2 \\ & & 3 & 0 & -12 & -42 & -114 \\ \hline & 1 & 0 & -4 & -14 & -38 & -112 \end{array}\)\(\text{Use synthetic substitution with }x=3\text{ and coefficients }1,-3,-4,-2,4,2.\)\(\text{The final number is the value of the function.}\)\(f(3)=-112\)
\(\textbf{3)}\) Find \( f(3) \) for \( f(x)=4x^4+2x^2-x-3\)
\(\text{The answer is }f(3)=336\)
\(\begin{array}{r|rrrrr} 3 & 4 & 0 & 2 & -1 & -3 \\ & & 12 & 36 & 114 & 339 \\ \hline & 4 & 12 & 38 & 113 & 336 \end{array}\)\(\text{Use synthetic substitution with }x=3\text{ and coefficients }4,0,2,-1,-3.\)\(\text{The final number is the value of the function.}\)\(f(3)=336\)
\(\textbf{4)}\) Find \( f(1) \) for \( f(x)=2x^4 – 3x^3 + x^2 + 2x – 5\)
\(\text{The answer is }f(1)=-3\)
\(\begin{array}{r|rrrrr} 1 & 2 & -3 & 1 & 2 & -5 \\ & & 2 & -1 & 0 & 2 \\ \hline & 2 & -1 & 0 & 2 & -3 \end{array}\)\(\text{Use synthetic substitution with }x=1\text{ and coefficients }2,-3,1,2,-5.\)\(\text{The final number is the value of the function.}\)\(f(1)=-3\)
\(\textbf{5)}\) Find \( f(-2) \) for \( f(x)=x^4 + 3x^3 – 2x + 4\)
\(\text{The answer is }f(-2)=0\)
\(\begin{array}{r|rrrrr} -2 & 1 & 3 & 0 & -2 & 4 \\ & & -2 & -2 & 4 & -4 \\ \hline & 1 & 1 & -2 & 2 & 0 \end{array}\)\(\text{Use synthetic substitution with }x=-2\text{ and coefficients }1,3,0,-2,4.\)\(\text{The final number is the value of the function.}\)\(f(-2)=0\)
\(\textbf{6)}\) Find \( f(2) \) for \( f(x)=3x^4 – 5x^2 + 6x – 1\)
\(\text{The answer is }f(2)=39\)
\(\begin{array}{r|rrrrr} 2 & 3 & 0 & -5 & 6 & -1 \\ & & 6 & 12 & 14 & 40 \\ \hline & 3 & 6 & 7 & 20 & 39 \end{array}\)\(\text{Use synthetic substitution with }x=2\text{ and coefficients }3,0,-5,6,-1.\)\(\text{The final number is the value of the function.}\)\(f(2)=39\)
\(\textbf{7)}\) Find \( f(-3) \) for \( f(x)=x^4 + 2x^3 – 5x – 8\)
\(\text{The answer is }f(-3)=34\)
\(\begin{array}{r|rrrrr} -3 & 1 & 2 & 0 & -5 & -8 \\ & & -3 & 3 & -9 & 42 \\ \hline & 1 & -1 & 3 & -14 & 34 \end{array}\)\(\text{Use synthetic substitution with }x=-3\text{ and coefficients }1,2,0,-5,-8.\)\(\text{The final number is the value of the function.}\)\(f(-3)=34\)
\(\textbf{8)}\) Find \( f(4) \) for \( f(x)=5x^4 – 2x^3 + 3x – 9\)
\(\text{The answer is }f(4)=1155\)
\(\begin{array}{r|rrrrr} 4 & 5 & -2 & 0 & 3 & -9 \\ & & 20 & 72 & 288 & 1164 \\ \hline & 5 & 18 & 72 & 291 & 1155 \end{array}\)\(\text{Use synthetic substitution with }x=4\text{ and coefficients }5,-2,0,3,-9.\)\(\text{The final number is the value of the function.}\)\(f(4)=1155\)
\(\textbf{9)}\) Find \( f(-1) \) for \( f(x)=2x^4 – 3x^3 + 4x^2 – 2x + 1\)
\(\text{The answer is }f(-1)=12\)
\(\begin{array}{r|rrrrr} -1 & 2 & -3 & 4 & -2 & 1 \\ & & -2 & 5 & -9 & 11 \\ \hline & 2 & -5 & 9 & -11 & 12 \end{array}\)\(\text{Use synthetic substitution with }x=-1\text{ and coefficients }2,-3,4,-2,1.\)\(\text{The final number is the value of the function.}\)\(f(-1)=12\)
\(\textbf{10)}\) Find \( f(5) \) for \( f(x)=3x^4 – 2x^2 + x – 7\)
\(\text{The answer is }f(5)=1823\)
\(\begin{array}{r|rrrrr} 5 & 3 & 0 & -2 & 1 & -7 \\ & & 15 & 75 & 365 & 1830 \\ \hline & 3 & 15 & 73 & 366 & 1823 \end{array}\)\(\text{Use synthetic substitution with }x=5\text{ and coefficients }3,0,-2,1,-7.\)\(\text{The final number is the value of the function.}\)\(f(5)=1823\)
\(\textbf{11)}\) Find \( f(-2) \) for \( f(x)=2x^4 – x^3 + 5x – 6\)
\(\text{The answer is }f(-2)=24\)
\(\begin{array}{r|rrrrr} -2 & 2 & -1 & 0 & 5 & -6 \\ & & -4 & 10 & -20 & 30 \\ \hline & 2 & -5 & 10 & -15 & 24 \end{array}\)\(\text{Use synthetic substitution with }x=-2\text{ and coefficients }2,-1,0,5,-6.\)\(\text{The final number is the value of the function.}\)\(f(-2)=24\)
\(\textbf{12)}\) Find \( f(2) \) for \( f(x)=x^3 – 4x^2 + 6x – 3\)
\(\text{The answer is }f(2)=1\)
\(\begin{array}{r|rrrr} 2 & 1 & -4 & 6 & -3 \\ & & 2 & -4 & 4 \\ \hline & 1 & -2 & 2 & 1 \end{array}\)\(\text{Use synthetic substitution with }x=2\text{ and coefficients }1,-4,6,-3.\)\(\text{The final number is the value of the function.}\)\(f(2)=1\)
\(\textbf{13)}\) Find \( f(-1) \) for \( f(x)=3x^4 – 2x^2 + 7\)
\(\text{The answer is }f(-1)=8\)
\(\begin{array}{r|rrrrr} -1 & 3 & 0 & -2 & 0 & 7 \\ & & -3 & 3 & -1 & 1 \\ \hline & 3 & -3 & 1 & -1 & 8 \end{array}\)\(\text{Use synthetic substitution with }x=-1\text{ and coefficients }3,0,-2,0,7.\)\(\text{The final number is the value of the function.}\)\(f(-1)=8\)
\(\textbf{14)}\) Find \( f(4) \) for \( f(x)=x^4 – 5x^2 + 4\)
\(\text{The answer is }f(4)=180\)
\(\begin{array}{r|rrrrr} 4 & 1 & 0 & -5 & 0 & 4 \\ & & 4 & 16 & 44 & 176 \\ \hline & 1 & 4 & 11 & 44 & 180 \end{array}\)\(\text{Use synthetic substitution with }x=4\text{ and coefficients }1,0,-5,0,4.\)\(\text{The final number is the value of the function.}\)\(f(4)=180\)
\(\textbf{15)}\) Find \( f(-3) \) for \( f(x)=2x^4 + 5x^3 – x^2 – 9\)
\(\text{The answer is }f(-3)=9\)
\(\begin{array}{r|rrrrr} -3 & 2 & 5 & -1 & 0 & -9 \\ & & -6 & 3 & -6 & 18 \\ \hline & 2 & -1 & 2 & -6 & 9 \end{array}\)\(\text{Use synthetic substitution with }x=-3\text{ and coefficients }2,5,-1,0,-9.\)\(\text{The final number is the value of the function.}\)\(f(-3)=9\)
\(\textbf{16)}\) Find \( f(1) \) for \( f(x)=4x^4 – 6x^3 + 3x – 10\)
\(\text{The answer is }f(1)=-9\)
\(\begin{array}{r|rrrrr} 1 & 4 & -6 & 0 & 3 & -10 \\ & & 4 & -2 & -2 & 1 \\ \hline & 4 & -2 & -2 & 1 & -9 \end{array}\)\(\text{Use synthetic substitution with }x=1\text{ and coefficients }4,-6,0,3,-10.\)\(\text{The final number is the value of the function.}\)\(f(1)=-9\)
\(\textbf{17)}\) Find \( f(3) \) for \( f(x)=x^5 + 2x^4 – 7x^3 + 6x – 4\)
\(\text{The answer is }f(3)=230\)
\(\begin{array}{r|rrrrrr} 3 & 1 & 2 & -7 & 0 & 6 & -4 \\ & & 3 & 15 & 24 & 72 & 234 \\ \hline & 1 & 5 & 8 & 24 & 78 & 230 \end{array}\)\(\text{Use synthetic substitution with }x=3\text{ and coefficients }1,2,-7,0,6,-4.\)\(\text{The final number is the value of the function.}\)\(f(3)=230\)
\(\textbf{18)}\) Find \( f(-2) \) for \( f(x)=5x^4 + 3x^2 – 4x + 1\)
\(\text{The answer is }f(-2)=101\)
\(\begin{array}{r|rrrrr} -2 & 5 & 0 & 3 & -4 & 1 \\ & & -10 & 20 & -46 & 100 \\ \hline & 5 & -10 & 23 & -50 & 101 \end{array}\)\(\text{Use synthetic substitution with }x=-2\text{ and coefficients }5,0,3,-4,1.\)\(\text{The final number is the value of the function.}\)\(f(-2)=101\)
\(\textbf{19)}\) Find \( f(0) \) for \( f(x)=7x^4 – 2x^3 + 5x^2 – x + 8\)
\(\text{The answer is }f(0)=8\)
\(\begin{array}{r|rrrrr} 0 & 7 & -2 & 5 & -1 & 8 \\ & & 0 & 0 & 0 & 0 \\ \hline & 7 & -2 & 5 & -1 & 8 \end{array}\)\(\text{Use synthetic substitution with }x=0\text{ and coefficients }7,-2,5,-1,8.\)\(\text{The final number is the value of the function.}\)\(f(0)=8\)
\(\textbf{20)}\) Find \( f(-4) \) for \( f(x)=x^4 – x^3 – 6x^2 + 2x + 8\)
\(\text{The answer is }f(-4)=224\)
\(\begin{array}{r|rrrrr} -4 & 1 & -1 & -6 & 2 & 8 \\ & & -4 & 20 & -56 & 216 \\ \hline & 1 & -5 & 14 & -54 & 224 \end{array}\)\(\text{Use synthetic substitution with }x=-4\text{ and coefficients }1,-1,-6,2,8.\)\(\text{The final number is the value of the function.}\)\(f(-4)=224\)
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