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Polynomials

Long Answer Type

431. If α, β are the zeroes of quadratic polynomial 2x² – 3x – 5, form a quadratic polynomial whose zeroes are α² and β².

397 Views

432. If α, β are the zeroes of quadratic poly-nomial x² – 3x + 2, form a quadratic polynomial whose roots are –α and –β.

161 Views

433. If α, β are the zeroes of quadratic polynomial x² – 3x – 1, form a quadratic polynomial whose roots are $1 over straight alpha squared comma space 1 over straight beta squared.$

Since, α, β are the zeroes of quadratic-polynomial x² – 3x – 1, then

straight alpha plus straight beta equals fraction numerator negative straight b over denominator straight a end fraction equals 3 over 1 equals 3 space and space straight alpha. space straight beta space equals space straight c over straight a equals fraction numerator negative 1 over denominator 1 end fraction equals negative 1

Now,

sum space left parenthesis straight alpha plus straight beta right parenthesis space equals space 1 over straight alpha squared plus 1 over straight beta squared

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fraction numerator left parenthesis 3 right parenthesis squared minus 2 left parenthesis negative 1 right parenthesis over denominator left parenthesis negative 1 right parenthesis to the power of 21 end fraction equals fraction numerator 9 plus 2 over denominator 1 end fraction equals 11

Product left parenthesis straight alpha. straight beta right parenthesis space equals space 1 over straight alpha squared cross times 1 over straight beta squared space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 1 over denominator left parenthesis αβ right parenthesis squared end fraction equals fraction numerator 1 over denominator left parenthesis negative 1 right parenthesis squared end fraction equals 1 over 1 equals 1

So, required quadratic polynomial be

space space space straight K open curly brackets straight x squared minus left parenthesis straight alpha plus straight beta right parenthesis straight x plus straight alpha. space straight beta close curly brackets

straight K open curly brackets straight x squared minus 11 straight x plus 1 close curly brackets.

122 Views

434. Factorize 2x⁴ + x³ – 14x² – 19x – 6 by x² + 3x + 2 and verify the division algorithm.

163 Views

Short Answer Type

435. Find all the zeroes of the polynomial x³ + 3x² – 2x – 6, if two of its zeroes are –

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1188 Views

436. Find all the zeroes of the polynomial 2a³ + x² – 6x – 3, if two of its zeroes are -

square root of 3 space and space square root of 3.

230 Views

437. If the polynomial 6x⁴+8x³ + 17x² + 21x + 7 is divided by another polynomial 3x²+ 4x +1, the remainder comes out to be (ax + b), find a and b.

2132 Views

438. If the polynomial x³ + 2x³ + 8x² + 12x + 18 is divided by another polynomial x² + 5, the remainder comes out to be px + q. Find the values of p and q.

211 Views

Long Answer Type

439. Find all the zeroes of 2x⁴ – 9x³ + 5x² + 3x– 1, if two of its zeroes are

2 plus square root of 3 space and space 2 minus square root of 3.

21071 Views

440. Find all the zeros of the polynomial x⁴ + x³ – 34x² – 4x + 120, if two of its zeroes are 2 and (–2).

2997 Views

Previous Year Papers

Test Series

Test Yourself

Long Answer Type

433. If α, β are the zeroes of quadratic polynomial x2 – 3x – 1, form a quadratic polynomial whose roots are

Short Answer Type

Long Answer Type

433. If α, β are the zeroes of quadratic polynomial x² – 3x – 1, form a quadratic polynomial whose roots are $1 over straight alpha squared comma space 1 over straight beta squared.$