Let ta be an integer such that all the real roots of the polynomial 2x⁵ + 5x⁴ + 10x³ + 10x² + 10x + 10 lie in the interval (a, a +1). Then $| a |$ is equal to……….

4 Views | Posted 2 weeks ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 weeks ago

Given $f (x) = 2 x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 10 x + 10$

$f (- 1) = 3 > 0 & f (- 2) = - 34 < 0$

So at least one root will lie in (-2, -1)

now $f' (x) = 10 x^{4} + 20 x^{3} + 30 x^{2} + 20 x + 10$

$= 10 [x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1]$

$= 10 x^{2} {(x + \frac{1}{x} + 1)}^{2} > 0 \forall x \in R$

So, f(x) be purely increasing function so exactly one root of f(x) that will lie in (-2, 1). Hence |a| = 2

Given <math> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mn>0</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mn>0</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mn>0</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mn>0</mn> </mrow> </math>  <math> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>3</mn> <mo>></mo> <mn>0</mn> <mo>&</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mn>4</mn> <mo><</mo> <mn>0</mn> </mrow> </math>           So at least one root will lie in (-2, -1)now <math> <mrow> <mi>f</mi> <mo>'</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mn>0</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mn>0</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mn>0</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mn>0</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mn>0</mn> </mrow> </math>   <math> <mrow> <mo>=</mo> <mn>1</mn> <mn>0</mn> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mrow> </math>             <math> <mrow> <mo>=</mo> <mn>1</mn> <mn>0</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>0</mn> <mo>∀</mo> <mi>x</mi> <mo>∈</mo> <mi>R</mi> </mrow> </math>           So, f(x) be purely increasing function so exactly one root of f(x) that will lie in (-2, 1). Hence |a| = 2

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
688k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

Let ta be an integer such that all the real roots of the polynomial 2x⁵ + 5x⁴ + 10x³ + 10x² + 10x + 10 lie in the interval (a, a +1). Then $| a |$ is equal to……….

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let ta be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a +1). Then | a | is equal to……….

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

Let ta be an integer such that all the real roots of the polynomial 2x⁵ + 5x⁴ + 10x³ + 10x² + 10x + 10 lie in the interval (a, a +1). Then $| a |$ is equal to……….