Advertisements
Advertisements
Question
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
t3 – 2t2 – 15t
Advertisements
Solution
t3 – 2t2 – 15t
Taking t common, we get,
t(t2 – 2t – 15)
Splitting the middle term of the equation t2 – 2t – 15, we get,
t(t2 – 5t + 3t – 15)
Taking the common factors out, we get,
t(t(t – 5) + 3(t – 5)
On grouping, we get,
t(t + 3)(t – 5)
So, the zeroes are,
t = 0
t + 3 = 0
`\implies` t = – 3
t – 5 = 0
`\implies` t = 5
Therefore, zeroes are 0, 5 and – 3
Verification:
Sum of the zeroes = – (coefficient of x2) ÷ coefficient of x3
α + β + γ = `- b/a`
(0) + (– 3) + (5) = `- (-2)/1`
= 2
Sum of the products of two zeroes at a time = coefficient of x ÷ coefficient of x3
αβ + βγ + αγ = `c/a`
(0)(– 3) + (– 3)(5) + (0)(5) = `-15/1`
= – 15
Product of all the zeroes = – (constant term) ÷ coefficient of x3
αβγ = `- d/a`
(0)(– 3)(5) = 0
= 0
APPEARS IN
RELATED QUESTIONS
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
4s2 – 4s + 1
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
4, 1
If α and β are the zeroes of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/(aalpha+b)+1/(abeta+b)`.
If the squared difference of the zeros of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the value of p.
If If α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3, find a polynomial whose roots are α + 2, β + 2.
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and product of its zeros as 3, −1 and −3 respectively.
Find the zeroes of the quadratic polynomial `(8x^2 ˗ 4)` and verify the relation between the zeroes and the coefficients
Find the quadratic polynomial whose zeroes are `2/3` and `-1/4`. Verify the relation between the coefficients and the zeroes of the polynomial.
Find a cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time and the product of its zeroes as 5, -2 and -24 respectively.
The product of the zeros of x3 + 4x2 + x − 6 is
If x + 2 is a factor of x2 + ax + 2b and a + b = 4, then
Case Study -1
The figure given alongside shows the path of a diver, when she takes a jump from the diving board. Clearly it is a parabola.
Annie was standing on a diving board, 48 feet above the water level. She took a dive into the pool. Her height (in feet) above the water level at any time ‘t’ in seconds is given by the polynomial h(t) such that h(t) = -16t2 + 8t + k.
The zeroes of the polynomial r(t) = -12t2 + (k - 3)t + 48 are negative of each other. Then k is ______.
If one of the zeroes of the quadratic polynomial (k – 1)x2 + k x + 1 is –3, then the value of k is ______.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`2s^2 - (1 + 2sqrt(2))s + sqrt(2)`
Given that `sqrt(2)` is a zero of the cubic polynomial `6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2)`, find its other two zeroes.
Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x and verify the relationship between the zeroes and the coefficients.
If the zeroes of the polynomial x2 + px + q are double in value to the zeroes of the polynomial 2x2 – 5x – 3, then find the values of p and q.
A quadratic polynomial whose sum and product of zeroes are 2 and – 1 respectively is ______.
