Question

8. If each zero of the polynomial x2 + mx + n is three times to the zeroes of 3x2 -5x + 2, find the
values of m and n.
​

Answer 1

SOLUTION

GIVEN

If each zero of the polynomial x² + mx + n is three times to the zeroes of 3x² - 5x + 2

TO DETERMINE

The value of m and n

EVALUATION

Here the given polynomial is 3x² - 5x + 2

Now we find the roots of the polynomial

For Zero of the polynomial 3x² - 5x + 2 we have

$\displaystyle \sf{3 {x}^{2} - 5x + 2 = 0 }$

$\displaystyle \sf{ \implies \: 3 {x}^{2} - 3x - 2x + 2 = 0 }$

$\displaystyle \sf{ \implies \: 3x(x - 1) - 2(x - 1) = 0 \: }$

$\displaystyle \sf{ \implies \: (x - 1) (3x - 2) = 0 \: }$

$\displaystyle \sf{ (x - 1) = 0 \: \: gives \: \: x = 1}$

$\displaystyle \sf{ (3x - 2) = 0 \: \: gives \: \: x = \frac{2}{3} }$

Therefore roots of the quadratic polynomial 3x² - 5x + 2 are $\displaystyle \sf{ 1 \: \: and \: \: \frac{2}{3} }$

Now it is given that each zero of the polynomial x² + mx + n is three times to the zeroes of 3x² - 5x + 2

Therefore roots of the quadratic polynomial x² + mx + n are $\displaystyle \sf{ 3 \: \: and \: \: 2}$

Now the quadratic polynomial whose roots are 3 , 2 are

$\sf{ {x}^{2} -(Sum \: of \: the \: zeroes )x + Product \: of \: the \: zeroes }$

$= \sf{ {x}^{2} - (3 + 2)x + (3 \times 2) }$

$= \sf{ {x}^{2} - 5x + 6 }$

Now Comparing with x² + mx + n we get

m = - 5 & n = 6

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