12.03.2020

# What is Irrational Root Theorem? Explain and give examples. 0

24.06.2023, solved by verified expert

if the irrational sum of a plus the square root of b is the root of a polynomial with rational coefficients, then a minus the square root of b, which is also an irrational number, is also a root of that polynomial.

Step-by-step explanation: Look it up

### Faq

Mathematics
Fernando’s response is incorrect because he inappropriately applied the Rational Root Theorem. Dennis’ response is incorrect. According to the Fundamental Theorem of Algebra, the polynomial p(x) cannot have six roots, or zeros, because it is only of degree 3. Emily’s response is correct because she correctly factored the polynomial, and correctly used the definition of zeros to reach her answer.

Step-by-step explanation:

The Rational Root Theorem offers a list of possible rational roots. Each needs to be tested to see if it is an actual rational root. Fernando and Dennis made inappropriate assumptions about what the Rational Root Theorem allowed them to conclude.

Mathematics
Fernando’s response is incorrect because he inappropriately applied the Rational Root Theorem. Dennis’ response is incorrect. According to the Fundamental Theorem of Algebra, the polynomial p(x) cannot have six roots, or zeros, because it is only of degree 3. Emily’s response is correct because she correctly factored the polynomial, and correctly used the definition of zeros to reach her answer.

Step-by-step explanation:

The Rational Root Theorem offers a list of possible rational roots. Each needs to be tested to see if it is an actual rational root. Fernando and Dennis made inappropriate assumptions about what the Rational Root Theorem allowed them to conclude.

Mathematics

if the irrational sum of a plus the square root of b is the root of a polynomial with rational coefficients, then a minus the square root of b, which is also an irrational number, is also a root of that polynomial.

Step-by-step explanation: Look it up

Mathematics

Step-by-step explanation:

A number is said to be a zero of a polynomial if when the number is substituted into the function the result is zero. That is if a is a zero of polynomial f(x), therefore f(a) = 0.

Since  P(−1)=0 P(0)=1 P(2+√3)=0, therefore -1 and 2+√3 are zeros of the polynomial.

Gary is right because there are 2 known zeros of P(x) which are −1 and 2+√3. Also 2 - √3 is also a root. From irrational root theorem, irrational roots are in conjugate pairs i.e. if a+√b is a root, a-√b is also a root.

Heather is not correct because if P(0) = 1, it means that 0 is not a root. It does not mean that 1 is a zero of P(x)

Irene is correct. since P(−1) and P(2+3–√) equal 0, 2 zeros of P(x) are −1 and 2+√3. They may be other zeros of P(x), but there isn't enough information to determine any other zeros of P(x)

Mathematics
1)
f(x) = 4x^2 − 25
f(x) = 4x^2 − 25 =0 implies  4x²=25, and x²=25/4, so the possible roots are
x=sqrt(25/4= 5/2   or x= -sqrt(25/4)= -5/2

2)
g(x) = 4x^2 + 25=0 implies  4x²= -25 and since  i²= -1 (complex number)
we can write 4x²= -25= 25i² so the possible root is x= sqrt(25i²/4)=5i/2 or x= -sqrt(25i²/4)=-5i/2
the answer is A. No, this polynomial has complex roots
3)
h(x) = 3x^2 − 25=0 implies  3x²=25, x²=25/3 so the possible roots are x= -sqrt(25/3)= -5/sqrt3 or x=sqrt(25/3)=5/sqrt3
the answer is B. No, this polynomial has irrational roots
Mathematics
1)
f(x) = 4x^2 − 25
f(x) = 4x^2 − 25 =0 implies  4x²=25, and x²=25/4, so the possible roots are
x=sqrt(25/4= 5/2   or x= -sqrt(25/4)= -5/2

2)
g(x) = 4x^2 + 25=0 implies  4x²= -25 and since  i²= -1 (complex number)
we can write 4x²= -25= 25i² so the possible root is x= sqrt(25i²/4)=5i/2 or x= -sqrt(25i²/4)=-5i/2
the answer is A. No, this polynomial has complex roots
3)
h(x) = 3x^2 − 25=0 implies  3x²=25, x²=25/3 so the possible roots are x= -sqrt(25/3)= -5/sqrt3 or x=sqrt(25/3)=5/sqrt3
the answer is B. No, this polynomial has irrational roots
Mathematics

if 2 + √3 is an irrational root to a polynomial, then its irrational conjugate 2- √3 is also.

Step-by-step explanation:

Given : A polynomial function, f(x), with rational coefficients has roots of –2 and square root of 3.

To find :  which of the following must also be a root of the function?

Solution : We have root of  f(x)  = 2 +√3.

By the  irrational conjugates theorem : it state that  states that if   a + √b is an irrational root to a polynomial, then its irrational conjugate a - √b is also

root.

Therefore, if 2 + √3 is an irrational root to a polynomial, then its irrational conjugate 2- √3 is also.

Mathematics

if 2 + √3 is an irrational root to a polynomial, then its irrational conjugate 2- √3 is also.

Step-by-step explanation:

Given : A polynomial function, f(x), with rational coefficients has roots of –2 and square root of 3.

To find :  which of the following must also be a root of the function?

Solution : We have root of  f(x)  = 2 +√3.

By the  irrational conjugates theorem : it state that  states that if   a + √b is an irrational root to a polynomial, then its irrational conjugate a - √b is also

root.

Therefore, if 2 + √3 is an irrational root to a polynomial, then its irrational conjugate 2- √3 is also.

Mathematics

A rational number is any number that can be written in the form, , where and are integers and .

We can rewrite .

Therefore option A is a rational number.

Option B is obviously a rational number because it is already in the form .

Option C cannot be written in the form because the decimal place does not repeat or recur and it does not terminate also.

Therefore is not a rational number.

As for option D, the decimal places recurs or repeats and it does not terminate. We can rewrite in the form, . hence it is a rational number.

Yes is a perfect square.

All numbers whose square roots are perfect squares are rational numbers.

If we raise we get .

In order words if we take the square root of we get a rational number.

That is Let the length from the wall to the base of the ladder be m.

The from Pythagoras Theorem, This implies that, We add the additive inverse of to both sides to obtain,  We take the positive square root of both sides to obtain,  .

We wan to estimate .

The highest perfect square that can be found in is .

We rewrite to obtain, .

We now split the square root sign to obtain, . . . .

The statement, every rational number is q square root is false.

We only need at least a counterexample to show that, the above statement is false.

Let be any real number.

Then , where and are integers.

This implies that Base on this final equation, can only be an integer if is a perfect number. Hence not every rational number is a square root because some numbers aren't perfect squares.

To find the translation vector that maps the blue rectangle on the red rectangle, we draw a vector connecting any two corresponding points as shown in the diagram.

The vector has horizontal component of and a vertical component of .

Therefore the mapping is . The correct answer is D.

Figure A accurately represents the Pythagorean Theorem because This implies that We can see that the hypotenuse square is equal to the sum of the squares of the lengths of the  two shorter legs.

Recall that, are Pythagorean triples. Mathematics

SI=(P*R*T)/100

P=2000

R=1.5

T=6

SI=(2000*1.5*6)/100

=(2000*9)/100

=180

Neil will earn interest of 180

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