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**Unformatted text preview: **ecompose −12x2 − 8x + 4 into a multiple of (2x − 3) plus a
constant term. Since we have found such a way, we can be sure it is the only way. The next example pulls together all of the concepts discussed in this section. 202 Polynomial Functions Example 3.2.2. Let p(x) = 2x3 − 5x + 3.
1. Find p(−2) using The Remainder Theorem. Check your answer by substitution.
2. Use the fact that x = 1 is a zero of p to factor p(x) and ﬁnd all of the real zeros of p.
Solution.
1. The Remainder Theorem states p(−2) is the remainder when p(x) is divided by x − (−2).
We set up our synthetic division tableau below. We are careful to record the coeﬃcient of x2
as 0, and proceed as above.
−2 2
0 −5
↓ −4
8
2 −4
3 3
−6
−3 According to the Remainder Theorem, p(−2) = −3. We can check this by direct substitution
into the formula for p(x): p(−2) = 2(−2)3 − 5(−2) + 3 = −16 + 10 + 3 = −3.
2. The Factor Theorem tells us that since x = 1 is a zero of p, x − 1 is a factor of p(x). To factor
p(x), we divide
1 2 0 ...

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