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# polynomial

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Consider the polynomial f(x) = 1 - 12x + 3x^2 - 4x^3 + 5x^4 and g(x) = 3 - 2x - 6x^3 + 19x^4. Find c such that the polynomial f(x) + cg(x) has degree 3.

May 22, 2021

All we need to do is choose c so that the 4th-degree terms of both polnomials are cancelled; whatever remains would then have degree 3. The fourth-degree terms of f and g are $$5{x}^{4}, and\ 19{x}^{4}$$, respectively. $$5{x}^{4}+(\frac{-5}{19})19{x}^{4}=5{x}^{4}-5{x}^{4}=0$$, so$$f(x)+(\frac{-5}{19})g(x)$$ would have degree 3.