LANA S.
asked • 11/06/23i need help solving this
The possible rational zeros of P(x) = 14x3 + 5x2 − 18x − 34
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1 Expert Answer
The possible rational zeros of this cubic polynomial are given by:
(±p)/(±q)
where p is a factor of the constant term (-34 in this case), and q is a factor of the leading coefficient (14 in this case).
± any nember of this set { 1, 1/2, 1/7, 1/14, 2, 2/7, 17, 17/2, 17/7, 17/14, 34, 34/7}
This is an application of the rational root theorem.
By the way, it turns out that there are no rational roots, one real irrational root, and one pair of complex conjugate roots.
LANA S.
can i get a little more explanation please?
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11/06/23
James S.
tutor
The rational root theorem states that any rational root (solution) must be of the form +/- p/q where p is a factor of the constant term and q is a factor of the leading coefficient. The constant term is 34. Its factors are 1, 2, 17, 34, -1, -2, -17, and -34. The leading coefficient of this polynomial is 14. Its factors are 1, 2, 7, 14, -1, -2, -7, and -14. When you combine each factor pair, one from the constant term list divided by one from the leading coefficient list, you generate the list I gave you in brackets. Any repeats are discarded so that you have only ine copy of each number.
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11/07/23
James S.
tutor
Then you check each possible root to see if it is an actual root. Graphing is a quick way to see that none of these potential roots are actual roots. Since the graph only crosses the x-axis once, the other 2 roots must be complex solutions. A polynomial of degree 3 (a cubic) always has 3 roots. The real root is irrational and cannot be written as a fraction. Does that clear up all of your questions?
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11/07/23
James S.
tutor
A rational number is a ratio of two integers. An irrational number cannot be expressed that way. Pi, the square root of two, and infinitely many more numbers are irrational.
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11/07/23
James S.
tutor
Rational numbers, when expressed as decimal fractions, either repeat (such as 1/3 =0.3333...) to infinity or terminate (such as 1/2 =0.5). Irrational numbers go to infinity without ever repeating.
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11/07/23
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Mark M.
Do you want a solution or just the possible rational roots?11/06/23