please give a step by step explanation for this answer
Answers
Answer:
-2, -1, -1/2, 3
Step-by-step explanation:
Given, p(x) = 2x⁴ + x³ - 14x² - 19x - 6.
(i) Since x = -2 is a factor, x + 2 is a factor.
(ii) Since x = -1 is a factor, x + 1 is a factor.
∴ Hence (x + 2)(x + 1) is a factor
= x² + x + 2x + 1
= x² + 3x + 1
∴ By dividing the polynomial by (x² + 3x + 1), we can find the other factors.
Long Division Method:
x² + 3x + 1) 2x⁴ + x³ - 14x² - 19x - 6 (2x² - 5x - 3
2x⁴ + 6x³ + 4x²
-----------------------------------
-5x³ - 18x² - 19x
-5x³ - 15x² - 10x
------------------------------------
-3x² - 9x - 6
-3x² - 9x - 6
--------------------------------------
0.
Now,
Factorize 2x² - 5x - 3.
⇒ 2x² - 6x + x - 3 = 0
⇒ 2x(x - 3) + (x - 3) = 0
⇒ x = -1/2, 3
Therefore, the zeroes of p(x) are -2, -1, -1/2, 3.
Hope it helps!
P(x) = 2 x⁴ + x³ - 14 x² - 19 x - 6
The number of sign changes in the coefficients : of P(x) = 1
the number of sign changes in the coefficients of P(-x) = 3
There are atmost 3 negative real roots and 1 positive real root.
two roots or zeros are : -2 and -1.
So (x+2) and (x+1) are the factors of P(x).
P(x) = (x² + 3 x + 2) (a x² + b x + c) = 2 x⁴ + x³ - 14 x² - 19 x - 6
comparing both sides we can say that:
a = 2 and c = - 3.
the coefficient of x³ on both sides are : 3 a + b = 1
b = -5
The roots of : 2 x² - 5 x - 3 = 0
(2 x + 1 ) ( x - 3 ) = 0
x = - 1/2 or 3