is x+b a factor of x^9+5b^2x^7+5bx^8-b^9
a little synthetic division shows that it is not.
However, it is a factor of
x^9+5b^2x^7+5bx^8+b^9
To determine if x+b is a factor of x^9+5b^2x^7+5bx^8-b^9, we can use the factor theorem. According to the factor theorem, if a polynomial f(x) has a factor (x-a), then f(a) = 0.
To apply the factor theorem in this case, substitute x = -b into the polynomial x^9+5b^2x^7+5bx^8-b^9 and check if the result is 0.
Replace x with -b:
(-b)^9 + 5b^2(-b)^7 + 5b(-b)^8 - b^9
Simplifying,
-b^9 + 5b^2(-b^7) + 5b(-b^8) - b^9
= -b^9 - 5b^9 - 5b^9 - b^9
= -12b^9
Since the result is not zero for x = -b, we can conclude that x+b is not a factor of x^9+5b^2x^7+5bx^8-b^9.
To determine whether x+b is a factor of x^9 + 5b^2x^7 + 5bx^8 - b^9, we can use polynomial division.
Step 1: Arrange the polynomial in descending order of powers:
x^9 + 5bx^8 + 5b^2x^7 - b^9
Step 2: Perform polynomial division by dividing each term of the dividend by the divisor x+b.
x^8 -b^2x^7 +b^3x^6
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x+b | x^9 + 5bx^8 + 5b^2x^7 - b^9
- (x^9 + bx^8)
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4bx^8 + 5b^2x^7
- (4bx^8 + b^2x^7)
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4b^2x^7 - b^9
Step 3: Keep dividing until the remainder no longer has a term with x. In this case, we have:
x^8 - b^2x^7 + b^3x^6 + ...
Since we encountered a remainder of 4b^2x^7 - b^9, which contains terms with x, x+b is not a factor of x^9 + 5b^2x^7 + 5bx^8 - b^9.