CBSE Class 10 Answered

find other zeroes of a polynomial -++ if two of its zeroes are 2 ans 3.

Asked by Akansha | 14 Jun, 2014, 06:33: AM

Expert Answer

Please check your question. The question should be as follows:

Find other zeroes of a polynomial x⁴-20x³+23x²+5x-6 if two of its zeroes are 2 and 3.

Solution: Let the other roots of the polynomial be $alpha$ and $beta$ .

Sum of the zeroes

$2 plus 3 plus alpha plus beta equals minus fraction numerator minus 20 over denominator 1 end fraction 5 plus alpha plus beta equals 20 alpha plus beta equals 20 minus 5 alpha plus beta equals 15 number space number space number space number space number space... left parenthesis i right parenthesis$

Product of the zeroes:

$2 cross times 3 cross times alpha cross times beta equals minus 6 6 alpha beta equals minus 6 alpha beta equals fraction numerator minus 6 over denominator 6 end fraction alpha beta equals minus 1 number space number space number space number space number space... left parenthesis i i right parenthesis$

Now,

$open parentheses alpha plus beta close parentheses squared equals alpha squared plus 2 alpha beta plus beta squared open parentheses alpha plus beta close parentheses squared equals alpha squared minus 2 alpha beta plus beta squared plus 4 alpha beta open parentheses alpha plus beta close parentheses squared equals open parentheses alpha minus beta close parentheses squared plus 4 alpha beta open parentheses alpha minus beta close parentheses squared equals open parentheses alpha plus beta close parentheses squared minus 4 alpha beta$

You are given the values of $alpha beta equals minus 1$ and $alpha plus beta equals 15$

Substitute the given values into the formula.

$open parentheses alpha minus beta close parentheses squared equals open parentheses alpha plus beta close parentheses squared minus 4 alpha beta open parentheses alpha minus beta close parentheses squared equals open parentheses 15 close parentheses squared minus 4 left parenthesis minus 1 right parenthesis open parentheses alpha minus beta close parentheses squared equals 225 plus 4 open parentheses alpha minus beta close parentheses squared equals 229 open parentheses alpha minus beta close parentheses equals square root of 229 number space number space number space number space number space... left parenthesis i i i right parenthesis$

Add equation (i) and (iii).

$alpha plus beta equals 15 bottom enclose alpha minus beta equals square root of 229 end enclose 2 alpha equals 15 plus square root of 229 alpha equals 1 half left square bracket 15 plus square root of 229 right square bracket$

Subtract equation (iii) from (i).

$alpha plus beta equals 15 bottom enclose alpha presubscript left parenthesis minus right parenthesis end presubscript minus beta presubscript left parenthesis minus right parenthesis end presubscript equals scriptbase square root of 229 end scriptbase presubscript minus end enclose 2 beta equals 15 minus square root of 229 beta equals 1 half left square bracket 15 minus square root of 229 right square bracket$

Therefore, the other zeroes of the polynomial are $1 half left square bracket 15 plus square root of 229 right square bracket$ and $1 half left square bracket 15 minus square root of 229 right square bracket$ .