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Part A. Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. Let $a$ be a constant. What is the largest possible degree of $f(x) + a\cdot g(x)$?

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

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Answer:

  A.  4

  B.  1

Step-by-step explanation:

The degree of a one-variable polynomial is the largest exponent of the variable.

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A.

For f(x) = x^4 -3x^2 +2 and g(x) = 2x^4 -6x^2 +2x -1, the sum f(x) +a·g(x) will be ...

  (x^4 -3x^2 +2) +a(2x^4 -6x^2 +2x -1)

  = (1 +2a)x^4 +(-3-6a)x^2 +2ax -a

The term with the largest exponent is (1 +2a)x^4, which has degree 4. This term will be non-zero for a ≠ -1/2.

The largest possible degree of f+ag is 4.

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B.

The polynomial sum is ...

  f+bg = (1 +2b)x^4 +(-3-6b)x^2 +2bx -b

When b = -1/2, the first two terms disappear and the sum becomes ...

  f+bg = -x +1/2 . . . . . . a polynomial of degree 1

The smallest possible degree of f+bg is 1.

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