- Substitute the real roots that we found for the equation stated in Prob. 3, one by one,
in place of x in the polynomial equation derived in the solution to Prob. 3. Verify that
these three roots work in that equation. - Multiply out the general binomial-trinomial equation to get an equation in polynomial
standard form for a cubic:
(a 1 x+b 1 )(a 2 x^2 +b 2 x+c)= 0- How can we tell from the coefficients and the constants alone how many real roots the
equation stated in Prob. 5 has (before multiplying it out)? - One of the “challenges” in the text required that we find the roots of the following
cubic equation in binomial-trinomial form:
(3x+ 5)(16x^2 − 56 x+ 49) = 0We found that the real roots are
x=−5/3 or x= 7/4Substitute these roots into the original equation, and go through the arithmetic to verify
that they’re accurate. Consider the following cubic equation in polynomial standard
form:
− 9 x^3 + 21 x^2 + 104 x+ 80 = 0- By means of synthetic division, verify that x= 5 is a real root of this equation.
- Using the quadratic formula on the results of the synthetic division performed in the
solution to Prob. 8, find the other root or roots of the original cubic, if any exist. Then
state the solution set. - The final “challenge” in the text revealed that x=−3/2 is the only real root of the
following cubic as expressed in the binomial-trinomial form:
(x+ 3/2)(6x^2 + 4 x+ 2) = 0How many real roots will the cubic have if the coefficient of x in the trinomial is
changed from 4 to −4? What will the new real roots be, if there are any?
Practice Exercises 431