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Class 10 Class 12

Polynomials

Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficient.
3x²–x–4

Solution not provided.
Ans. $negative 1 comma space 4 over 3$

94 Views

If the polynomial 6x⁴ + 8x³ – 5x² + ax + b is exactly divisible by the polynomial 2x² – 5, then find the values of a and b.

If $space straight p left parenthesis straight x right parenthesis space equals space 6 straight x to the power of 4 plus 8 straight x cubed minus 5 straight x squared plus ax plus straight b space straight q left parenthesis straight x right parenthesis space equals space 2 straight x squared space minus space 5$
If Now, and
Now,
$20 straight x plus ax plus straight b minus 25 equals 0 straight x left parenthesis 20 plus straight a right parenthesis plus straight b minus 25 equals 0. straight x plus 0 rightwards double arrow space space space space space space space space space straight a space plus space 20 space equals space 0$

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$rightwards double arrow$ $straight a equals negative 20$
and $straight b equals 25$
Hence, other zeroes are -20, 25.

243 Views

Find all the zeros of the polynomial 2x⁴ + 7x³ – 19x² – 14x + 30, if two of its zeros are

square root of 2 space and space minus square root of 2

Since, two zeros are

square root of 2 space and space square root of 2 comma

∴

left parenthesis straight x minus square root of 2 right parenthesis space left parenthesis straight x plus square root of 2 right parenthesis space equals space straight x squared minus 2

is a factor of the given polynomial.
Now, we divide the given polynomial by x² – 2.
Since, two zeros are ∴ is a factor of the given polynomial.Now

∴

2 straight x to the power of 4 plus 7 straight x cubed minus 19 straight x squared minus 14 straight x plus 30 space equals space left parenthesis straight x squared minus 2 right parenthesis space left parenthesis 2 straight x squared plus 7 straight x minus 15 right parenthesis

Now,2x² + 7x – 15
= 2x² + 10x –3x – 15
= 2y (x + 5) – 3 (x + 5) = (x + 5) (2x – 3)
So. its zeros are given by x = – 5 and x = 3/2 . Hence, the zeros of the given polynomials are :

square root of 2 comma space left parenthesis negative square root of 2 right parenthesis comma space left parenthesis negative 5 right parenthesis space and space 3 over 2

Ans.

3131 Views

Find all the zeroes of the polynomial

negative 2 square root of 2 straight x cubed minus 18 straight x squared plus 24 square root of 2 straight x plus 72

given that

2 square root of 3 space straight x to the power of 4 space and space minus 2 square root of 3

are two of its zeroes.

Since two zeroes are $2 square root of 3 space and space minus 2 square root of 3$
$rightwards double arrow$ $left parenthesis straight x minus 2 square root of 3 right parenthesis space left parenthesis straight x plus 2 square root of 3 right parenthesis space equals space straight x squared minus 12$ is a factor of the given polynomial.
Now,
Since two zeroes are is a factor of the given polynomia
Now, $straight x to the power of 4 minus 2 square root of 2 straight x cubed minus 18 straight x squared plus 24 square root of 2 straight x plus 72$
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$equals left parenthesis straight x plus 2 square root of 3 right parenthesis space left parenthesis straight x minus 2 square root of 3 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets straight x left parenthesis straight x minus 3 square root of 2 right parenthesis space plus space square root of 2 left parenthesis straight x minus 3 square root of 2 right parenthesis close square brackets$

= $left parenthesis straight x plus 2 square root of 3 right parenthesis space left parenthesis straight x minus 2 square root of 3 right parenthesis space left parenthesis straight x plus square root of 2 right parenthesis space left parenthesis straight x minus 3 square root of 2 right parenthesis$
Hence, other zeroes are
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222 Views

Find all other zeroes of $straight x to the power of 4 plus straight x cubed minus 23 straight x squared minus 3 straight x plus 60 comma$ if it is given that two of its zeroes are $square root of 3 space and space minus square root of 3.$

Since two zeroes are $square root of 3 space and space minus square root of 3$
$rightwards double arrow$ $left parenthesis straight x minus square root of 3 right parenthesis space left parenthesis straight x plus square root of 3 right parenthesis space equals space straight x squared minus 3$ is a factor of the given polynomial. Now,
Since two zeroes are is a factor of