1. 55 Next Problem
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Solution
Consider the following polynomials:
What is the sum of all of the coefficients of ?
A(x)
B(x)
C(x)
= −x + 12 + 4 − 8x2
x4
= −6 + 8 + 1x2
x3
= 9 − 3 − 1 + 4x.x4
x3
A(x) + B(x) + 2C(x)
Solution 1: The sum of coefficients of the polynomial can be obtained by evaluating at 1. Hence, we seek the value of
.
Solution 2: Rearranging the terms of each polynomial by the degree, we get
Adding in terms with 0 coefficient, we get
Thus, . Therefore, the sum of all the coefficients is .
A(1) + B(1) + 2C(1) = [−1 + 12 + 4 − 8] + [−6 + 8 + 1] + 2 × [9 − 3 − 1 + 4] = 28
A(x)
B(x)
C(x)
= −8 + 12 − x + 4x4
x2
= 8 − 6 + 1x3
x2
= 9 − 3 + 4x − 1.x4
x3
A(x)
B(x)
2C(x)
= −8x4
= 0x4
= 18x4
+0x3
+8x3
−6x3
+12x2
−6x2
+0x2
−x
+0x
+8x
+4
+1
−2.
A(x) + B(x) + 2C(x) = 10 + 2 + 6 + 7x + 3x4
x3
x2
10 + 2 + 6 + 7 + 3 = 28
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