Question 54069This question is from textbook Algebra and Trigonometry
: Please help me with this problem:
The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.
This question is from textbook Algebra and Trigonometry
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.
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Since the coefficients are real, and since 3+i is a root, 3-1 must be
a root.
so the polynomial has factors: (x-4),(x-(3+i)),(x-(3-i))
f(x) is the product of these three factors.
f(x)= (x-4)((x-3)-i)((x-3)+i)
f(x)= (x-4)((x-3)^2-i^2)
f(x)= (x-4)(x^2-6x+9+1}
f(x)= (x-4)(x^2-6x+10)
f(x)= x^3-10x^2+34x+10
Cheers,
Stan H.
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