Consider the followingPx x4 x3 5x2 25x 750 Find all the z

Consider the following.P(x) = x⁴ + x³ 5x² + 25x 750

Find all the zeros of the polynomial function. (Hint: First determine the rational zeros. Enter your answers as a comma-separated list. Enter all answers including repetitions.)

x =   

Write the polynomial as a product of its leading coefficient and its linear factors.

P(x) =

Consider the following. P(x) = 3x⁵ + 4x⁴ + 5x³ + 6x² 2x 4

Find all the zeros of the polynomial function. (Hint: First determine the rational zeros. Enter your answers as a comma-separated list. Enter all answers including repetitions.)

x =   

Write the polynomial as a product of its leading coefficient and its linear factors.

P(x) =

Solution

P(x) = x^4 + x^3 5x^2 + 25x 750

Use rational root theorem :

he factor of the leading coefficient (1) is 1 .The factors of the constant term (-750) are 1 2 3 5 6 10 15 25 30 50 75 125 150 250 375 750 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±2/1, ±3/1, ±5/1, ±6/1, ±10/1, ±15/1, ±25/1, ±30/1, ±50/1, ±75/1, ±125/1, ±150/1, ±250/1, ±375/1, ±750/1

Substitute the possible roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

Use Factor Theorem :

( x^4 + x^3 5x^2 + 25x 750)/(x -5) =x^3 +6x^2 +25x +150

Solve : x^3 +6x^2 +25x +150

If we plug these values into the polynomial P(x), we obtain P(5)=0.

The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction pq, where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factor of the leading coefficient (1) is 1 .The factors of the constant term (150) are 1 2 3 5 6 10 15 25 30 50 75 150 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±2/1, ±3/1, ±5/1, ±6/1, ±10/1, ±15/1, ±25/1, ±30/1, ±50/1, ±75/1, ±150/1

Substitute the possible roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

If we plug these values into the polynomial P(x), we obtain P(6)=0.

( x^3 +6x^2 +25x +150)/ (x +6) = x^2 +25

solve x^2 +25 ---> x = +5i , -5i

So, roots of polynomial are x = 5 , -6 , 5i , -5i

P(x) = (x -5)(x+6)(x^2 +25)