Question

Given the geometric series: 5×(3)⁴+5(3)³+5(3)²explain why the series converges​

Answer 1

Answer:

$45*(9x^2+4)$

Step-by-step explanation:

STEP

1

:

Equation at the end of step

1

:

((5x2•(34))+(5•(33)))+(5•32)

STEP

2

:

Equation at the end of step 2

((5x2•(34))+(5•(33)))+(5•32)

STEP

3

:

Equation at the end of step

3

:

((5x2 • (34)) + (5 • 33)) + (5•32)

STEP

4

:

Equation at the end of step 4

((5x2 • (34)) + (5•33)) + (5•32)

STEP

5

:

Equation at the end of step

5

:

((5x2 • 34) + (5•33)) + (5•32)

STEP

6

:

Equation at the end of step 6

((5•34x2) + (5•33)) + (5•32)

STEP

7

:

STEP

8

:

Pulling out like terms

8.1 Pull out like factors :

405x2 + 180 = 45 • (9x2 + 4)

Polynomial Roots Calculator :

8.2 Find roots (zeroes) of : F(x) = 9x2 + 4

Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 9 and the Trailing Constant is 4.

The factor(s) are:

of the Leading Coefficient : 1,3 ,9

of the Trailing Constant : 1 ,2 ,4

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 13.00

-1 3 -0.33 5.00

-1 9 -0.11 4.11

-2 1 -2.00 40.00

-2 3 -0.67 8.00

Note - For tidiness, printing of 13 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

45 • (9x2 + 4)