Algebra Know-It-ALL
We multiply this sum by the “test root” and place the product beneath the entry −7: (^210) − 7 − 6 24# 12## We add the numbers i ...
426 Cubic Equations in Real Numbers 410 − 7 − 6 ### 1### 410 − 7 − 6 4## 1### 410 − 7 − 6 4## 14## (^410) − 7 − 6 416# 14## (^41 ...
310 − 7 − 6 ### #### 310 − 7 − 6 ### 1### 310 − 7 − 6 3## 1### (^310) − 7 − 6 3## 13## (^310) − 7 − 6 39# 13## (^310) − 7 − 6 39 ...
428 Cubic Equations in Real Numbers What are the other roots? We can write the binomial-trinomial form of a cubic equation strai ...
−3/2 6138 3 ### #### Subsequent steps proceed as follows: −3/2 6138 3 ### 6### −3/2 6138 3 − 9 ## 6### −3/2 6138 3 − 9 ## 64## − ...
430 Cubic Equations in Real Numbers That’s (x+ 3/2). In the trinomial factor, the coefficient of x^2 is 6, the coefficient of x ...
Substitute the real roots that we found for the equation stated in Prob. 3, one by one, in place of x in the polynomial equatio ...
432 CHAPTER 26 Polynomial Equations in Real Numbers In this chapter, we’ll examine some ways to look for real roots of higher-de ...
(− 7 x− 12)^7 = 0 (− 118 x+ 59)^13 = 0 (− 3 x)^17 = 0 In the last case, the stand-alone constant is 0. Multiplying out You can m ...
434 Polynomial Equations in Real Numbers If we substitute 3/2 for x here, we get (2× 3/2 − 3)(2 × 3/2 − 3)(2 × 3/2 − 3)(2 × 3/2 ...
We now have a binomial to the nth equation with n= 4. We can find the solitary real root by setting the binomial equal to 0: x+ ...
436 Polynomial Equations in Real Numbers What are the real roots? As we search for real roots to a higher-degree equation in bin ...
We have a root in every single case where one of these factors becomes equal to 0. There are not four such instances here, but 1 ...
438 Polynomial Equations in Real Numbers Polynomial Standard Form A higher-degree equation in polynomial standard form contains ...
Digging for Real Roots Now that we’ve learned how to recognize a polynomial equation, it’s time to think about finding the real ...
440 Polynomial Equations in Real Numbers Finding an upper bound How can we find an upper bound for the roots of the polynomial e ...
24 1 − 2 − 13 14 24 24 528 # # 1 22 515 # # 24 1 − 2 − 13 14 24 24 528 12,360 # 1 22 515 # # 24 1 − 2 − 13 14 24 24 528 12,360 # ...
442 Polynomial Equations in Real Numbers − 24 1 − 2 − 13 14 24 #### 1#### − 24 1 − 2 − 13 14 24 − 24 ### 1#### − 24 1 − 2 − 13 1 ...
The numbers in the last row alternate in sign, telling us that −24 is a lower bound for the real solution set. The fact that the ...
444 Polynomial Equations in Real Numbers With synthetic division, check every r to see if it is a root of the original polynomi ...
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