+389 Suppose the polynomial p(x)=x^3+ax^2+bc+c has the property that the mean of its zeroes, the product of its zeroes, and the sum of its coefficients are all equal. If the y-intercept of the graph of y=p(x) is 2, what is b?
+118608 Suppose the polynomial p(x)=x^3+ax^2+bc+c has the property that the mean of its zeroes, the product of its zeroes, and the sum of its coefficients are all equal. If the y-intercept of the graph of y=p(x) is 2, what is b?
for this answer I will assume it is a typo
The y intercept is the constant so c=2
\(p(x)=x^3+ax^2+bx+c\\ p(x)=x^3+ax^2+bx+2\\\)
\(\mbox{Let the zeros be }\;\alpha,\;\; \beta, \;\;and \;\; \gamma\\ \alpha+\beta+\gamma=\frac{-a}{1}\quad so \\ \frac {\alpha+\beta+\gamma}{3}=\frac{-a}{3}\\ \alpha\beta\gamma=\frac{-c}{1}=-2\\ \mbox{The coefficients are 1, a and b}\\ so\\ \frac {\alpha+\beta+\gamma}{3}=\alpha\beta\gamma=1+a+b\\ \frac{-a}{3}=-2=1+a+b\\ a=6\\~\\ -2=1+6+b\\ -2=7+b\\ b=-9\\ \)
+118608 Dabae,
Is it meant to be bc as you have entered OR is it meant to be bx ??
+118608 Suppose the polynomial p(x)=x^3+ax^2+bc+c has the property that the mean of its zeroes, the product of its zeroes, and the sum of its coefficients are all equal. If the y-intercept of the graph of y=p(x) is 2, what is b?
for this answer I will assume it is a typo
The y intercept is the constant so c=2
\(p(x)=x^3+ax^2+bx+c\\ p(x)=x^3+ax^2+bx+2\\\)
\(\mbox{Let the zeros be }\;\alpha,\;\; \beta, \;\;and \;\; \gamma\\ \alpha+\beta+\gamma=\frac{-a}{1}\quad so \\ \frac {\alpha+\beta+\gamma}{3}=\frac{-a}{3}\\ \alpha\beta\gamma=\frac{-c}{1}=-2\\ \mbox{The coefficients are 1, a and b}\\ so\\ \frac {\alpha+\beta+\gamma}{3}=\alpha\beta\gamma=1+a+b\\ \frac{-a}{3}=-2=1+a+b\\ a=6\\~\\ -2=1+6+b\\ -2=7+b\\ b=-9\\ \)
