When the polynomial f(x)=5x^4 + 4x^3 + 3x^2 + Px + Q is divided by x^2 - 1, the remainder is zero. Determine the value of P + Q
(x^2 - 1) ( a x^2 + b x + c) = 5x^4 + 4x^3 + 3 x^2 + Px + Q
x^2 ( a x^2 + b x + c) = ax^4 + b x^3 + c x^2
-1 ( a x^2 + b x + c) = 0x^4 + 0 x^3 - a x^2 - b x - c
add = a x^4 + b x^3 +(c-a) x^2 - b x - c
a = 5
b = 4
c - a = 3 so c = 8
P = -b = -4
Q = -c = -8
or, you can do a long division
or two synthetic divisions using +1 and -1
To determine the value of P + Q, we need to find the values of P and Q that make the remainder zero when the polynomial f(x) is divided by x^2 - 1.
When the polynomial f(x) is divided by x^2 - 1, the remainder can be found by polynomial long division. Let's perform the long division:
5x^2 + 9x + 3
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x^2 - 1 | 5x^4 + 4x^3 + 3x^2 + Px + Q
To eliminate the x^4 term, we can multiply x^2 - 1 by 5x^2, giving us 5x^4 - 5x^2. Subtracting this from the original polynomial, we have:
5x^2 + 9x + 3
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x^2 - 1 | 5x^4 + 4x^3 + 3x^2 + Px + Q
- (5x^4 - 5x^2)
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9x^2 + Px + Q
Now our polynomial becomes 9x^2 + Px + Q. We need to eliminate the x^2 term, so we multiply x^2 - 1 by 9x, giving us 9x^3 - 9x. Subtracting this from the previous result, we have:
(9x^3 - 9x) + 9x^2 + Px + Q
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x^2 - 1 | 5x^4 + 4x^3 + 3x^2 + Px + Q
- (5x^4 - 5x^2)
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9x^2 + Px + Q
- (9x^3 - 9x)
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(-9x^3 + 9x^2 + Px + Q)
We are left with (-9x^3 + 9x^2 + Px + Q). We can eliminate the x^3 term by multiplying x^2 - 1 by -9x, giving us -9x^3 + 9x. Subtracting this from the previous result, we have:
(-9x^3 + 9x) + (-9x^3 + 9x^2 + Px + Q)
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x^2 - 1 | 5x^4 + 4x^3 + 3x^2 + Px + Q
- (5x^4 - 5x^2)
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9x^2 + Px + Q
- (9x^3 - 9x)
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(-9x^3 + 9x^2 + Px + Q)
- (-9x^3 + 9x)
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Px + Q + 9x
Now we are left with (Px + Q + 9x). We need to eliminate the x term, so we multiply x^2 - 1 by P, giving us Px^2 - P. Subtracting this from the previous result, we have:
(Px^2 - P) + (Px + Q + 9x)
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x^2 - 1 | 5x^4 + 4x^3 + 3x^2 + Px + Q
- (5x^4 - 5x^2)
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9x^2 + Px + Q
- (9x^3 - 9x)
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(-9x^3 + 9x^2 + Px + Q)
- (-9x^3 + 9x)
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Px + Q + 9x
- (Px^2 - P)
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Q - Px^2 + (9 + P)x - P
The remainder we have obtained is Q - Px^2 + (9 + P)x - P. Since the remainder is zero, we know that Q - Px^2 + (9 + P)x - P = 0.
Now, equating the coefficients of like terms, we have:
- Px^2 + (9 + P)x - P = 0
This can be rewritten as:
-Px^2 + 9x + Px - P = 0
Combining like terms, we have:
(9 - P)x - (P - P)x^2 - P = 0
Again, combining like terms, we have:
(9 - P)x - P(x^2 - 1) = 0
Since we know that the remainder is zero, we can conclude that the coefficients of each term must be zero. Therefore:
9 - P = 0 => P = 9
P - P = 0 => P = 0
Simplifying this, we find that P = 9.
Now, substitute this value of P back into the equation:
(9 - P)x - P(x^2 - 1) = 0
(9 - 9)x - 9(x^2 - 1) = 0
0x - 9(x^2 - 1) = 0
-9(x^2 - 1) = 0
Distributing the -9, we have:
-9x^2 + 9 = 0
Add 9x^2 to both sides to isolate the variable:
9 = 9x^2
Divide both sides by 9:
1 = x^2
From this equation, we have x = ±1.
Now let's substitute the obtained values of x back into the original polynomial:
For x = 1:
f(1) = 5(1)^4 + 4(1)^3 + 3(1)^2 + P(1) + Q
= 5 + 4 + 3 + P + Q
= 12 + P + Q
For x = -1:
f(-1) = 5(-1)^4 + 4(-1)^3 + 3(-1)^2 + P(-1) + Q
= 5 - 4 + 3 + P - Q
= 4 + P - Q
Since the remainder is zero, both equations should equal to zero. Therefore, we can set up a system of equations:
12 + P + Q = 0
4 + P - Q = 0
Subtracting the second equation from the first, we have:
(12 + P + Q) - (4 + P - Q) = 0
12 + P + Q - 4 - P + Q = 0
8 + 2Q = 0
2Q = -8
Q = -4
Now that we have the values of P and Q, we can calculate P + Q:
P + Q = 9 + (-4) = 5.
Therefore, P + Q = 5.