Advertisements
Advertisements
प्रश्न
If 1 and –2 are two zeroes of the polynomial (x3 – 4x2 – 7x + 10), find its third zero.
Advertisements
उत्तर
Let f(x) = x3 – 4x2 – 7x + 10
Since 1 and –2 are the zeroes of f(x), it follows that each one of (x – 1) and (x + 2) is a factor of f(x).
Consequently, (x – 1) (x + 2) = (x2 + x – 2) is a factor of f(x).
On dividing f(x) by (x2 + x – 2), we get:
`x^2 + x - 2")"overline(x^3 - 4x^2 - 7x + 10)"("x - 5`
x3 + x2 – 2x
– – +
–5x2 – 5x + 10
–5x2 – 5x + 10
+ + –
X
f(x) = 0 ⇒ (x2 + x – 2) (x – 5) = 0
⇒ (x – 1) (x + 2) (x – 5) = 0
⇒ x = 1 or x = –2 or x = 5
Hence, the third zero is 5.
APPEARS IN
संबंधित प्रश्न
if α and β are the zeros of ax2 + bx + c, a ≠ 0 then verify the relation between zeros and its cofficients
Verify that the numbers given along side of the cubic polynomials are their zeroes. Also verify the relationship between the zeroes and the coefficients.
`2x^3 + x^2 – 5x + 2 ; 1/2, 1, – 2`
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`0, sqrt5`
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
1, 1
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
`p(x) = x^2 + 2sqrt2x + 6`
Find the zeroes of the quadratic polynomial 4x2 – 4x + 1 and verify the relation between the zeroes and the coefficients.
Verify that 5, –2 and `1/3` are the zeroes of the cubic polynomial p(x) = (3x3 – 10x2 – 27x + 10) and verify the relation between its zeros and coefficients.
The product of the zeros of x3 + 4x2 + x − 6 is
Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial ______.
If 2 and `1/2` are the zeros of px2 + 5x + r, then ______.
The below picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.
If the sum of the roots is –p and the product of the roots is `-1/"p"`, then the quadratic polynomial is:
An asana is a body posture, originally and still a general term for a sitting meditation pose, and later extended in hatha yoga and modern yoga as exercise, to any type of pose or position, adding reclining, standing, inverted, twisting, and balancing poses. In the figure, one can observe that poses can be related to representation of quadratic polynomial.
The zeroes of the quadratic polynomial `4sqrt3"x"^2 + 5"x" - 2sqrt3` are:
If one of the zeroes of the quadratic polynomial (k – 1)x2 + k x + 1 is –3, then the value of k is ______.
If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
3x2 + 4x – 4
For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
`21/8, 5/16`
If α and β are the zeros of a polynomial f(x) = px2 – 2x + 3p and α + β = αβ, then p is ______.
Find the sum and product of the roots of the quadratic equation 2x2 – 9x + 4 = 0.
If p(x) = x2 + 5x + 6, then p(– 2) is ______.
A quadratic polynomial whose sum and product of zeroes are 2 and – 1 respectively is ______.
