JJkitty787
Answered

Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Ask your questions and receive precise answers from experienced professionals across different disciplines. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

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 $f(x) + b\cdot g(x)$?

Sagot :

sqdancefan

Answer:

  A.  4

  B.  1

Step-by-step explanation:

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

__

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.

__

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.

Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.