Find the values of c and d if x^2-7x+5 is a factor of cx^3 + dx^2 +25.
To find the values of c and d, we can use polynomial long division to divide the polynomial cx^3 + dx^2 + 25 by the factor x^2 - 7x + 5.
First, let's set up the long division:
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x^2 - 7x + 5 | cx^3 + dx^2 + 25
To begin the long division, divide the first term in the polynomial being divided (cx^3) by the first term in the divisor (x^2). This gives us cx.
cx
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x^2 - 7x + 5 | cx^3 + dx^2 + 25
Multiply the divisor (x^2 - 7x + 5) by cx, and write the result below the dividend:
cx
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x^2 - 7x + 5 | cx^3 + dx^2 + 25
- cx^3 + 7cx^2 - 5cx
Subtract this result from the dividend by changing the signs and combine like terms:
cx
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x^2 - 7x + 5 | cx^3 + dx^2 + 25
- cx^3 + 7cx^2 - 5cx
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+ (d - 7c)x^2 + 5cx + 25
Next, repeat the process by dividing the first term of the new dividend ((d - 7c)x^2) by the first term in the divisor (x^2). This gives us (d - 7c)x.
cx + (d - 7c)x
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x^2 - 7x + 5 | cx^3 + dx^2 + 25
- cx^3 + 7cx^2 - 5cx
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+ (d - 7c)x^2 + 5cx + 25
- (d - 7c)x^2 + 7(d - 7c)x
Subtract this result from the previous step:
cx + (d - 7c)x
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x^2 - 7x + 5 | cx^3 + dx^2 + 25
- cx^3 + 7cx^2 - 5cx
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+ (d - 7c)x^2 + 5cx + 25
- (d - 7c)x^2 + 7(d - 7c)x
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- 12(d - 7c)x + 25
We now have the new dividend -12(d - 7c)x + 25. Since this is a linear term, we cannot divide it by x^2 - 7x + 5 since the divisor is a quadratic factor. Therefore, for the divisor x^2 - 7x + 5 to be a factor of cx^3 + dx^2 + 25, the linear term in the result must be zero. Thus, we need to set -12(d - 7c)x + 25 equal to zero and solve for c and d.
-12(d - 7c)x + 25 = 0
-12(d - 7c) = 0
d - 7c = 0
d = 7c
Therefore, the values of c and d are related by d = 7c.