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What must be added to f (x) = 4 x 4 +2x3− 2 x 2 +x−1 so that the resulting polynomial is divisible by g (x) = x 2 + 2x −3 ?
- a)61x - 65
- b)-61x + 65
- c)-61x - 65
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
What must be added to f (x) = 4 x 4 +2x3− 2 x 2 +x−1 so th...
Here
x2 + 2 x — 3)4 x4 + 2 x3 — 2 x2 + x — l(4 x2 — 6 x + 22
x2 + 2 x — 3)4 x4 + 2 x3 — 2 x2 + x — l(4 x2 — 6 x + 22
Clearly -61x + 65 must be added.
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Community Answer
What must be added to f (x) = 4 x 4 +2x3− 2 x 2 +x−1 so th...
Explanation:
Step 1: Find the quotient when f(x) is divided by g(x)
- To find the quotient, divide f(x) by g(x) using polynomial division.
- Quotient = 4x^2 - 6x + 9
Step 2: Multiply the quotient by g(x) to get the product polynomial
- Multiply the quotient (4x^2 - 6x + 9) by g(x) (x^2 + 2x - 3) to get the product polynomial.
- Product polynomial = 4x^4 - 6x^3 + 9x^2 + 8x^3 - 12x^2 + 18x - 12x^2 + 18x - 27
- Product polynomial = 4x^4 + 2x^3 - 2x^2 + 27
Step 3: Find the difference between the product polynomial and f(x)
- Subtract f(x) from the product polynomial to find the difference.
- Difference = 4x^4 + 2x^3 - 2x^2 + 27 - (4x^4 + 2x^3 - 2x^2 + x - 1)
- Difference = 4x^4 + 2x^3 - 2x^2 + 27 - 4x^4 - 2x^3 + 2x^2 - x + 1
- Difference = -x + 28
Step 4: Determine the polynomial to add to f(x) to make it divisible by g(x)
- The polynomial that needs to be added to f(x) is the difference (-x + 28).
- This can be simplified to -x + 28 = -1x + 28 = -x + 28
Therefore, the polynomial that must be added to f(x) = 4x^4 + 2x^3 - 2x^2 + x - 1 is -x + 28 to make the resulting polynomial divisible by g(x) = x^2 + 2x - 3.
Thus, the correct answer is option -61x + 65.
Step 1: Find the quotient when f(x) is divided by g(x)
- To find the quotient, divide f(x) by g(x) using polynomial division.
- Quotient = 4x^2 - 6x + 9
Step 2: Multiply the quotient by g(x) to get the product polynomial
- Multiply the quotient (4x^2 - 6x + 9) by g(x) (x^2 + 2x - 3) to get the product polynomial.
- Product polynomial = 4x^4 - 6x^3 + 9x^2 + 8x^3 - 12x^2 + 18x - 12x^2 + 18x - 27
- Product polynomial = 4x^4 + 2x^3 - 2x^2 + 27
Step 3: Find the difference between the product polynomial and f(x)
- Subtract f(x) from the product polynomial to find the difference.
- Difference = 4x^4 + 2x^3 - 2x^2 + 27 - (4x^4 + 2x^3 - 2x^2 + x - 1)
- Difference = 4x^4 + 2x^3 - 2x^2 + 27 - 4x^4 - 2x^3 + 2x^2 - x + 1
- Difference = -x + 28
Step 4: Determine the polynomial to add to f(x) to make it divisible by g(x)
- The polynomial that needs to be added to f(x) is the difference (-x + 28).
- This can be simplified to -x + 28 = -1x + 28 = -x + 28
Therefore, the polynomial that must be added to f(x) = 4x^4 + 2x^3 - 2x^2 + x - 1 is -x + 28 to make the resulting polynomial divisible by g(x) = x^2 + 2x - 3.
Thus, the correct answer is option -61x + 65.
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What must be added to f (x) = 4 x 4 +2x3− 2 x 2 +x−1 so that the resulting polynomial is divisible by g (x) = x 2 + 2x −3 ?a)61x - 65b)-61x + 65c)-61x - 65d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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