Polynomial questions

[jashaan]

Hi there, could someone please help me out with these two multiple choice questions from the topic of polynomials:
1. If a > b > c and P(x) = (x−a)2(x−b)(x−c), then P(x) < 0 for x∈  (BTW the answer is D)

 A (−∞,a) B(−∞,b)  C(−∞,c) D (c,b) E (b,a)


2. The equation x3 + 5x−10 = 0 has only one solution. This solution lies between  (BTW, the answer is D)
A.  −2 and−1     B. −1 and 0    C. 0 and 1     D. 1 and 2     E. 2 and 8

Thank youuu and good luck for your exams <333

[S_R_K]

Hi there, could someone please help me out with these two multiple choice questions from the topic of polynomials:
1. If a > b > c and P(x) = (x−a)2(x−b)(x−c), then P(x) < 0 for x∈  (BTW the answer is D)

 A (−∞,a) B(−∞,b)  C(−∞,c) D (c,b) E (b,a)

Sketch a graph, and look where the graph is below the x-axis.

It is a quartic polynomial, with a positive x^4 term, so P(x) > 0 when x approaches positive or negative infinity.

There is an x-intercept at x = a, and (x – a)^2 is also a factor, so x = a is also a turning point.
There are also x-intercepts at x = b and x = c, but here the graph just cuts through the x-axis.

Putting all this together, the graph will be above the x-axis when x < a, it will be above the x-axis when a < x < b, it will be below the x-axis when b < x < c, and then it goes back above the x-axis when c < x. This gives D as the answer.

2. The equation x3 + 5x−10 = 0 has only one solution. This solution lies between  (BTW, the answer is D)
A.  −2 and−1     B. −1 and 0    C. 0 and 1     D. 1 and 2     E. 2 and 8

Thank youuu and good luck for your exams <333

Just do a CAS solve.

[me1431]

"2. The equation x3 + 5x−10 = 0 has only one solution. This solution lies between"
Just do a CAS solve.

but the question doesn't allow CAS calculator so how would you solve this on your own? thanks

[fun_jirachi]

but the question doesn't allow CAS calculator so how would you solve this on your own? thanks

For any continuous function \(f\), if \(f(a) < 0\) and \(f(b) > 0\) or vice versa, there must be at least one solution on the interval \((a, b)\). Here, \(f(1) < 0\) and \(f(2) > 0\). Given we're told only one solution exists, it must lie on this interval.