SOLUTION: Let $b$ be a constant. What is the smallest possible degree of the polynomial $f(x) + b\cdot g(x)$? Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Let $b$ be a constant. What is the smallest possible degree of the polynomial $f(x) + b\cdot g(x)$? Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1.      Log On


   

Question 1209336: Let $b$ be a constant. What is the smallest possible degree of the polynomial $f(x) + b\cdot g(x)$? Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1.
Answer by ikleyn(52782) About Me  (Show Source):
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Let $b$ be a constant. What is the smallest possible degree of the polynomial f(x) + b*g(x)?
Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1.
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Regularly, we expect that f(x) + b*g(x)  is the 4-degree polynomial.   But if you take b= -1/2,

the terms containing x^4 will cancel each other; the terms containing x^2 also will cancel each other, 

and then you will get the sum f(x) + b*g(x) as the polynomial of the degree 1 (one),

which is a linear binomial.  So, the smallest possible degree is 1 at b = -1/2.

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