If the zeroes of the polynomial $x^3 - 3x^2 + x + 1$ are $a-b, a, a + b$, find $a$ and $b$.
Given:
The zeroes of the polynomial $x^3 - 3x^2 + x + 1$ are $a-b, a, a + b$.
To do:
We have to find $a$ and $b$.
Solution:
Let $\alpha, \beta$ and $\gamma$ be the zeroes of polynomial $x^3 – 3x^2 + x + 1$.
This implies,
$\alpha = a-b, \beta = a$ and $\gamma = a + b$.
Therefore,
Sum of zeroes $= \alpha + \beta + \gamma$
$(a – b) + a + (a + b)=3$
$a-b + a + a + b = 3$
$3a = 3$
$a= 1$........…(i)
Product of zeroes $= \alpha \beta \gamma$
$(a – b) a (a + b) = -1$
$(a^2 – b^2)a = -1$
$a^3 – ab^2 = -1$...… (ii)
Putting the value of $a$ from equation (i) in equation (ii), we have,
$(1)^3-(1)b^2 = -1$
$1 – b^2 = -1$
$b^2 = 1 + 1$
$b^2 = 2$
$b = \pm \sqrt2$
Hence, $a = 1$ and $b = \pm \sqrt2$
Related Articles
- If the zeroes of the quadratic polynomial $x^2+( a+1)x+b$ are $2$ and $-3$, then $a=?,\ b=?$.
- If $x = 0$ and $x = -1$ are the roots of the polynomial $f(x) = 2x^3 - 3x^2 + ax + b$, find the value of $a$ and $b$.
- If \( x-\sqrt{3} \) is a factor of the polynomial \( a x^{2}+b x-3 \) and \( a+b=2-\sqrt{3} \). Find the values of \( a \) and \( b \).
- Given that the zeroes of the cubic polynomial \( x^{3}-6 x^{2}+3 x+10 \) are of the form \( a \), \( a+b, a+2 b \) for some real numbers \( a \) and \( b \), find the values of \( a \) and \( b \) as well as the zeroes of the given polynomial.
- Find the value of $(x-a)^3 + (x-b)^3 + (x-c)^3 - 3 (x-a)(x-b)(x-c)$ if $a+b+c = 3x$
- If \( 2 x+3 \) and \( x+2 \) are the factors of the polynomial \( g(x)=2 x^{3}+a x^{2}+27 x+b \), find the value of the constants $a$ and $b$.
- For which values of \( a \) and \( b \), are the zeroes of \( q(x)=x^{3}+2 x^{2}+a \) also the zeroes of the polynomial \( p(x)=x^{5}-x^{4}-4 x^{3}+3 x^{2}+3 x+b \) ? Which zeroes of \( p(x) \) are not the zeroes of \( q(x) \) ?
- If \( x+1 \) is a factor of \( 2 x^{3}+a x^{2}+2 b x+1 \), then find the values of \( a \) and \( b \) given that \( 2 a-3 b=4 \).
- Find the values of $a$ and $b$ so that $(x + 1)$ and $(x - 1)$ are factors of $x^4 + ax^3 - 3x^2 + 2x + b$.
- If both $x + 1$ and $x - 1$ are factors of $ax^3 + x^2 - 2x + b$, find the values of $a$ and $b$.
- Choose the correct answer from the given four options in the following questions:If the zeroes of the quadratic polynomial \( x^{2}+(a+1) x+b \) are 2 and \( -3 \), then(A) \( a=-7, b=-1 \)(B) \( a=5, b=-1 \)(C) \( a=2, b=-6 \)(D) \( a=0, b=-6 \)
- Simplify:(i) $2x^2 (x^3 - x) - 3x (x^4 + 2x) -2(x^4 - 3x^2)$(ii) $x^3y (x^2 - 2x) + 2xy (x^3 - x^4)$(iii) $3a^2 + 2 (a + 2) - 3a (2a + 1)$(iv) $x (x + 4) + 3x (2x^2 - 1) + 4x^2 + 4$(v) $a (b-c) - b (c - a) - c (a - b)$(vi) $a (b - c) + b (c - a) + c (a - b)$(vii) $4ab (a - b) - 6a^2 (b - b^2) -3b^2 (2a^2 - a) + 2ab (b-a)$(viii) $x^2 (x^2 + 1) - x^3 (x + 1) - x (x^3 - x)$(ix) $2a^2 + 3a (1 - 2a^3) + a (a + 1)$(x) $a^2 (2a - 1) + 3a + a^3 - 8$(xi) $\frac{3}{2}-x^2 (x^2 - 1) + \frac{1}{4}-x^2 (x^2 + x) - \frac{3}{4}x (x^3 - 1)$(xii) $a^2b (a - b^2) + ab^2 (4ab - 2a^2) - a^3b (1 - 2b)$(xiii) $a^2b (a^3 - a + 1) - ab (a^4 - 2a^2 + 2a) - b (a^3- a^2 -1)$.
- Choose the correct answer from the given four options in the following questions:If one of the zeroes of the cubic polynomial \( x^{3}+a x^{2}+b x+c \) is \( -1 \), then the product of the other two zeroes is(A) \( b-a+1 \)(B) \( b-a-1 \)(C) \( a-b+1 \)(D) \( a-b-1 \)
- Find the values of \( a \) and \( b \), if 2 and 3 are zeroes of $x^3+ax^2+bx-30$.
- $( x-2)$ is a common factor of $x^{3}-4 x^{2}+a x+b$ and $x^{3}-a x^{2}+b x+8$, then the values of $a$ and $b$ are respectively.
Kickstart Your Career
Get certified by completing the course
Get Started