Use synthetic division to find P(–3) for P(x)=x^4–2x^3–4x+4 .
To find P(-3) using synthetic division, we will divide the polynomial P(x) by (x + 3).
First, we need to set up the synthetic division table:
-3 | 1 -2 0 -4 4
Since the leading coefficient is 1, we write it at the top of the division table:
-3 | 1 -2 0 -4 4
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Next, we bring down the first coefficient, which is 1:
-3 | 1 -2 0 -4 4
| 1
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Next, we multiply the -3 (the divisor) by 1 (the number we brought down) and write the result below the next coefficient:
-3 | 1 -2 0 -4 4
| 1
| -3
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Next, we add (-2) and (-3) to get -5, and write the result below the next coefficient:
-3 | 1 -2 0 -4 4
| 1 -5
| -3
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Next, we multiply the -3 (the divisor) by -5 (the number we just calculated) and write the result below the next coefficient:
-3 | 1 -2 0 -4 4
| 1 -5 15
| -3
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Next, we add 0 and 15 to get 15, and write the result below the next coefficient:
-3 | 1 -2 0 -4 4
| 1 -5 15
| -3 15
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Lastly, we multiply the -3 (the divisor) by 15 (the number we just calculated) and write the result below the last coefficient:
-3 | 1 -2 0 -4 4
| 1 -5 15 -33
| -3 15
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Finally, we add -4 and -33 to get -37, which is the remainder:
P(-3) = -37
Therefore, P(-3) = -37.