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Polynomials

Short Answer Type

321. The lateral surface area of a cube is 4 times the square of its edge, find the edge of a cube whose lateral surface area is given by:

4 straight x squared minus 8 minus square root of 128 straight x

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322. Divide the polynomial 3x⁴ - 4x³ - 3x - 1 by x - 1 and find its quotient and remainder.

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323. Simplify:

fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis space plus space left parenthesis straight b squared minus straight c squared right parenthesis cubed plus left parenthesis straight c squared minus straight a squared right parenthesis cubed over denominator left parenthesis straight a minus straight b right parenthesis cubed plus left parenthesis straight b minus straight c right parenthesis cubed plus left parenthesis straight c minus straight a right parenthesis cubed end fraction

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324. If 3a - 2b + 5c = 5 and 6ab + 10bc - 15ac = 14, find the value of 27a³ + 125c³ + 90 abc -8b³.

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325. Find the value of (x - a)³ + (x - b)³ + (x - c)³ - 3(x - a) (x - b) (x - c) where a + b + c = 3x

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Long Answer Type

326. The graph of y = p(x) are given in the figure below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

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Short Answer Type

327. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
$straight x squared minus 2 straight x minus 8$

We have,
$space space space straight x squared minus 2 straight x minus 8 space equals space straight x squared minus 4 straight x plus 2 straight x minus 8$
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So, the value of x² – 2x - 8 is zero when x - 4 = 0 or x + 2 = 0.
i.e., when x = 4 or x = –2.
Therefore, the zeroes of x² – 2x – 8 are 4 and - 2. Now,
Sum of zeroes = (4) + (-2)
$equals 2 over 1 equals fraction numerator negative left parenthesis Coefficient space of space straight x right parenthesis over denominator Coefficient space of space straight x squared end fraction$
Product of zeroes = (4) (-2)
$equals fraction numerator negative 8 over denominator 1 end fraction equals fraction numerator Constant space term over denominator Coefficient space of space straight x squared end fraction.$