14.01.2020

Which is the polynomial function of lowest degree with rational real coefficients, a leading coefficient of 3 and roots startroot 5 endroot and 2?

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17.02.2022, solved by verified expert
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Which is the polynomial function of lowest degree, №15222927, 14.01.2020 01:22

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Which is the polynomial function of lowest degree, №15222927, 14.01.2020 01:22

Step-by-step explanation:

For a polynomial function of lowest degree with rational real coefficients, each root has multiplicity of 1.

The polynomial has roots Which is the polynomial function of lowest degree, №15222927, 14.01.2020 01:22 and 2 with a leading coefficient of 3.

By the irrational root theorem of polynomials, Which is the polynomial function of lowest degree, №15222927, 14.01.2020 01:22 is also a root of the required polynomial.

By the factor theorem, we can write the polynomial in factored form as:

Which is the polynomial function of lowest degree, №15222927, 14.01.2020 01:22

We expand, applying difference of two squares to obtain

Which is the polynomial function of lowest degree, №15222927, 14.01.2020 01:22

We expand further using the distributive property to get:

Which is the polynomial function of lowest degree, №15222927, 14.01.2020 01:22

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Mathematics
Step-by-step answer
P Answered by PhD

Since there are two zeroes, the polynomial function of lowest degree is a quadratic polynomial.

Let it be p(x).

So, p(x) = (x-\sqrt{5})(x-2)

Note that the leading coefficient should be 3.

Therefore, p(x) = 3(x-\sqrt{5} )(x-2)

=3[x^{2}-(2+\sqrt{5})x+2\sqrt{5}  ]

=3x^{2}-3(2+\sqrt{5})x+6\sqrt{5}

Mathematics
Step-by-step answer
P Answered by PhD

a) f(x)=3x^{3}-6x^{2}-15x+30

Step-by-step explanation:

1) In this question we've been given "a", the leading coefficient. and two roots:

x_{1}=\sqrt{5}\:x_{2}=2

2) There's a theorem, called the Irrational Theorem Root that states:

If one root is in this form x'=\sqrt{a}+b  then its conjugate x''=\sqrt{a}-b. is also a root of this polynomial.

Therefore

x_3=-\sqrt{5}

3) So, applying this Theorem we can rewrite the equation, by factoring. Remembering that x is the root. Since the question wants it in this expanded form then:

f(x)=3(x-\sqrt{5})(x+\sqrt{5})(x-2)\Rightarrow 3x^{3}-6x^{2}-15x+30

Mathematics
Step-by-step answer
P Answered by PhD

f(x) = 3x² -21x + 30

Step-by-step explanation:

Polynomial function of lowest degree with roots as 'a' and 'b' and leading coefficient as 'c' is given by:

P(x) = c (x - a) (x - b)

Given: Leading coefficient = 3, Roots = sqrt 5 and 2

P(x) = 3 (x - 5) (x - 2)

      = 3 ( x² - 2x - 5x +10)

       = 3 ( x² - 7x +10)

       = 3x² - 21x + 30

Mathematics
Step-by-step answer
P Answered by PhD

a) f(x)=3x^{3}-6x^{2}-15x+30

Step-by-step explanation:

1) In this question we've been given "a", the leading coefficient. and two roots:

x_{1}=\sqrt{5}\:x_{2}=2

2) There's a theorem, called the Irrational Theorem Root that states:

If one root is in this form x'=\sqrt{a}+b  then its conjugate x''=\sqrt{a}-b. is also a root of this polynomial.

Therefore

x_3=-\sqrt{5}

3) So, applying this Theorem we can rewrite the equation, by factoring. Remembering that x is the root. Since the question wants it in this expanded form then:

f(x)=3(x-\sqrt{5})(x+\sqrt{5})(x-2)\Rightarrow 3x^{3}-6x^{2}-15x+30

Mathematics
Step-by-step answer
P Answered by PhD
We are asked in the problem to determine the polynomial function with a leading coeffiient of 3 and whose roots are square root of 5 and 2. In this case, the poolynomial function becomes:

(x-√5) (x-2) = x^2 - 2(2+√5) + 2√5
since the leading coefficient is 3, then the answer becomes
(x^2 - 2(2+√5) + 2√5)*3 = 3x^2 - 6(2+√5) + 6√5
 
Mathematics
Step-by-step answer
P Answered by PhD

the polynomial is: 3x^2-21x+30

Step-by-step explanation:

we know that the polynomial function p(x) of lowest degree with roots as 'a' and 'b' and leading coefficient as 'c' is given by:p(x)=c(x-a)(x-b)

here we are given that the roots are 5 and 2 and the leading coefficient is 3.

so the polynomial p(x) of lowest degree with the above properties is: p(x)=3(x-5)(x-2)

p(x)=3(x^2-5x-2x+10)

p(x)=3(x^2-7x+10)

p(x)=3x^2-21x+30..

Mathematics
Step-by-step answer
P Answered by PhD

f(x) = 3x² -21x + 30

Step-by-step explanation:

Polynomial function of lowest degree with roots as 'a' and 'b' and leading coefficient as 'c' is given by:

P(x) = c (x - a) (x - b)

Given: Leading coefficient = 3, Roots = sqrt 5 and 2

P(x) = 3 (x - 5) (x - 2)

      = 3 ( x² - 2x - 5x +10)

       = 3 ( x² - 7x +10)

       = 3x² - 21x + 30

Mathematics
Step-by-step answer
P Answered by Specialist

f(x) = 3x^{3}  - 6x^{2} - 15x + 30

Step-by-step explanation:

If  \sqrt{5}  is a root, then so is -\sqrt{5}

f(x) = 3(x + \sqrt{5})(x - \sqrt{5})(x - 2)

     = 3(x^{2} - 5)(x - 2)

     = 3(x^{3} - 2x^{2} - 5x + 10)

     = 3x^{3}  - 6x^{2} - 15x + 30

Mathematics
Step-by-step answer
P Answered by PhD
The polynomial functions A) and C) have a leading coefficient of 3, but C) has a lower degree.
f ( x ) = 3 x² - 21 x + 30
3 ( x² - 7 x + 10 ) = 3 ( x² - 5 x - 2 x + 10 ) =
= 3 ( x ( x - 5 ) - 2 ( x - 5 )) = 3 ( x - 2 ) ( x - 5 ) = 0
x = 2 and x = 5 ( roots )

C ) f ( x ) = 3 x² - 21 x + 30
Mathematics
Step-by-step answer
P Answered by PhD

Since there are two zeroes, the polynomial function of lowest degree is a quadratic polynomial.

Let it be p(x).

So, p(x) = (x-\sqrt{5})(x-2)

Note that the leading coefficient should be 3.

Therefore, p(x) = 3(x-\sqrt{5} )(x-2)

=3[x^{2}-(2+\sqrt{5})x+2\sqrt{5}  ]

=3x^{2}-3(2+\sqrt{5})x+6\sqrt{5}

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