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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.

267 Views

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

108 Views

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

307 Views

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

118 Views

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.

2329 Views

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

931 Views

328. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
$4 straight s squared minus 4 straight s plus 1$

We have,
4s² – 4s + 1 = 4s² – 2s –2s + 1 = (2s–1)(2s– 1)
So, the value of 4s² – 4s + 1 is zero, when 2s – 1 = 0 or 2s – 1 = 0
i.e. when

straight s equals 1 half space or space straight s equals 1 half

Therefore, the zeroes of 4s² – 4s + 1 are

1 half space and space 1 half.

Now,
Sum of zeroes =

1 half plus 1 half

equals 1 over 1 equals fraction numerator negative left parenthesis Coefficient space of space straight s right parenthesis over denominator Coefficient space of space straight s squared end fraction

Product of zeroes =

1 half cross times 1 half equals 1 fourth

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791 Views

329. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

6 straight x squared minus 3 minus 7 straight x

862 Views

330. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

4 straight u squared plus 8 straight u

870 Views