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Answer:
1. (x+1) (x-7) (x+2) = 0 true.
x= -1 , 7 , -2
Step-by-step explanation:
1. f(x)= x³ - 4x² - 19x - 14
Set x³ - 4x² - 19x - 14 is equal to 0.
x³ - 4x² - 19x - 14 = 0
Solve for x:
Factor the left side of the equation.
Factor x³ - 4x² - 19x - 14 using the rational roots test.
If a polynomial function has integer coefficients, then every rational zero will have the form
p/q where p is a factor of the constant and q
where p is a factor of the constant and q is a factor of the leading coefficient.
p= ±1 , ±14 , ±2 , ±7
q= ±1
Find every combination of ± p/q . These are the possible roots of the polynomial function.
±1 , ±14 , ±2 , ±7
Substitute the -1 and simplify the expression In this case, the expression is equal to 0 so -1 is a root of the polynomial.
Substitute −1 into the polynomial. (−1)³ −4 (−1)² −19 . −1 −14
Raise -1 to the power of 3.
-1 -4 (-1)² -19. -1 -14
Raise -1 to the power of 2.
-1 -4 - 1 -19. -1 -14
Multiply -4 by 1
-1 -4 -19. -1 -14
Subtract 4 from -1.
-5 -19. -1 -14
Multiply -19 by 1.
-5 + 19 - 14
Add -5 and 19.
14 - 14
Subtract 14 from 14.
0
Since -1 is a known root, divide polynomial by
x+1 to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
x³–4x²–19x–14
x+1
Divide x³ -4x² - 19x - 14 by x+1.
x² -5x -14
Write x³ - 4x² -19x - 14 as a set of factors.
(x+1) (x²-5x-14) = 0
Factor x²-5x-14 using the AC method.
Consider the form of x²+bx+c . Find a pair of integers whose product is c and whose sum is b. In this case whose product is -14 and whose sum is -5,-7,2.
Write the factored form using these integers.
(x+1) ((x-7) (x+2))=0
Remove unnecessary parenthesis.
(x+1)(x-7)(x+2)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
x+1=0
x-7=0
x+2=0
Set the first factor equal to 0 and solve.
x=-1
Set the next factor equal to 0 and solve.
x-7=0
Add 7 to both sides of the equation.
x=7
Set the next factor equal to 0 and solve.
x+2=0
Subtract 2 from both sides of the equation.
x=2
The final solution is all the values that make (x+1)(x−7)(x+2)=0 true.
x=−1,7,−2