if p(x) is a quadratic function and p(l) =a, p(m) =b, p(n) =c then find p(x).
Share
if p(x) is a quadratic function and p(l) =a, p(m) =b, p(n) =c then find p(x).
Sign Up to our social questions and Answers Engine to ask questions, answer people’s questions, and connect with other people.
Login to our social questions & Answers Engine to ask questions answer people’s questions & connect with other people.
SOLUTION
GIVEN
TO DETERMINE
p(x)
EVALUATION
Here it is given that p(l) =a, p(m) =b, p(n) =c
Let us assume that
[tex] \sf{p(x) = u(x - l)(x - m) + v(x -m )(x -n ) + w(x - n)(x - l) } \: ...(1)[/tex]
Now p(l) = a gives from Equation (1)
[tex] \sf{p(l) = u(l- l)(l- m) + v(l -m )(l-n ) + w(l - n)(l - l) \: }[/tex]
[tex] \implies \sf{a= v(l -m )(l-n ) }[/tex]
[tex] \displaystyle \implies \sf{ v = \frac{a}{(l - m)(l - n)} }[/tex]
Again p(m) = b gives from Equation (1)
[tex] \sf{p(m) = u(m - l)(m - m) + v(m -m )(m -n ) + w(m - n)(m - l) }[/tex]
[tex] \implies \sf{b = w(m - n)(m - l) }[/tex]
[tex] \displaystyle \implies \sf{ w= \frac{b}{(m - n)(m- l)} }[/tex]
Again p(n) = b gives from Equation (1)
[tex] \sf{p(n) = u(n- l)(n - m) + v(n -m )(n-n ) + w(n - n)(n - l) } [/tex]
[tex] \implies \sf{c= u(n- l)(n - m)} [/tex]
[tex] \displaystyle \implies \sf{ u= \frac{c}{(n - l)(n- m)} }[/tex]
Now putting the values of u, v, w in Equation (1) we get
[tex] \displaystyle \sf{p(x) = c \frac{(x - l)(x - m)}{(n - l)(n- m)} + a \frac{(x -m )(x -n )}{( l-m )(l -n )} + b \frac{(x - n)(x - l)}{(m - n)(m - l)} } [/tex]
On rearranging we get
[tex] \displaystyle \sf{p(x) = a \frac{(x -m )(x -n )}{( l-m )(l -n )} + b \frac{(x - n)(x - l)}{(m - n)(m - l)} +c \frac{(x - l)(x - m)}{(n - l)(n- m)} } [/tex]
Which is the required polynomial
FINAL ANSWER
Hence the required polynomial is
[tex] \displaystyle \sf{p(x) = a \frac{(x -m )(x -n )}{( l-m )(l -n )} + b \frac{(x - n)(x - l)}{(m - n)(m - l)} +c \frac{(x - l)(x - m)}{(n - l)(n- m)} } [/tex]
━━━━━━━━━━━━━━━━
LEARN MORE FROM BRAINLY
Find a cubic function with the
given zeros -6, 7, -4
https://brainly.in/question/23346934