x-3 and x+3 are factors of 4x3+ax2 +bx find a and b.
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Given that (x-2) and (x+3) are factors of 2x^4-ax^3-10x^2+bx-54, find a and b.
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Student Answers
GIORGIANA1976 | STUDENT
We'll apply the division of polynomials. Considering the fact that x-2 and x+3 are factors, that means that 2 and -3 are the roots of the polynomial 2x^4-ax^3-10x^2+bx-54. We know that we could write a polynomial as a product of linear factors, depending on it's roots.
2x^4-ax^3-10x^2+bx-54=(x-2)(x+3)(cx^2+dx+e)
We've noticed that multiplying (x-2)(x+3), we'll obtain a second degree polynomial. But the given polynomial has the fourth degree, so we have to multiply (x-2)(x+3) with another polynomial of second degree.
We'll do the math, to the right side and the result will be:
(x-2)(x+3) = x^2+x-6
(x-2)(x+3)(cx^2+dx+e) = (x^2+x-6)(cx^2+dx+e)
(x^2+x-6)(cx^2+dx+e) = cx^4+dx^3+ex^2+cx^3+dx^2+ex-6cx^2-6dx-6e
If 2 polynomials are identically, that means that the corresponding coefficients are equal.
2x^4= cx^4, so c=2
-ax^3=(d+c)x^3, so d+c=-a, where c=2, d+2=-a
-10x^2=(e+d-6c)x^2
-10=e+d-6c
e+d=-10+12
e+d=2, where e=8, so d=2-8, d=-6
bx=(e-6d)x
b=e-6d
-6e=-54
e=8
d+2=-a, a=-2-d, where d=-6
a=-2-(-6)
a=6-2
a=4
b=e-6d, where d=-6 and e=8
b=8-6(-6)
b=8+36
b=44