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The polynomial x^2 - 4x + 3 is a factor of x^3+(a-4)x^2 + (3-4a)x + 3. Calculate what a is. ?
The polynomial x^2 - 4x + 3 is a factor of x^3+(a-4)x^2 + (3-4a)x + 3. Calculate what a is.
I just can't seem to solve it, i want to see the whole process of solving it.
Thank you in advance. :)
1 Answer
- Pi R SquaredLv 71 decade agoFavorite Answer
Hi,
"a" = 1 <==ANSWER
If x² - 4x + 3 is a factor of x³ + (a - 4)x² + (3 - 4a)x + 3, then in order to have x³ at the front of the polynomial, the other factor has to start with x and in order for the constant to be 3, the constant in the other factor must be a 1.
So (x + 1)(x² - 4x + 3) = x³ + (a - 4)x² + (3 - 4a)x + 3
Multiply out the left-hand side.
(x + 1)(x² - 4x + 3) = x³ + (a - 4)x² + (3 - 4a)x + 3
x³ - 4x² + 3x + x² - 4x + 3 = x³ + (a - 4)x² + (3 - 4a)x + 3
x³ - 3x² - x + 3 = x³ + (a - 4)x² + (3 - 4a)x + 3
This means from the x² terms that -3 = a - 4. This solves to a = 1. From the x terms -1 = 3 - 4a and -4 = -4a, so a = 1 again.
So "a" = 1 <==ANSWER
I hope that helps!! :-)
