B) -11
C) -7
Solve the rational inequality: $\frac{3 x+12}{x-6} \geq 0$
B) $(-\infty$,
$\cup(6, \infty)$
D) $(-\infty$,
Find the multiplicity of each zero of $p(x)=5 x^6(x+2)^7(x-3)^7$
Answer (2)
The zeros of the polynomial p ( x ) = 5 x 6 ( x + 2 ) 7 ( x − 3 ) 7 are x = 0 (multiplicity 6), x = − 2 (multiplicity 7), and x = 3 (multiplicity 7).
The critical points of the inequality x − 6 3 x + 12 ≥ 0 are x = − 4 and x = 6 .
The sign chart shows that the inequality is positive in the intervals ( − ∞ , − 4 ) and ( 6 , ∞ ) , and zero at x = − 4 .
The solution to the inequality is ( − ∞ , − 4 ] ∪ ( 6 , ∞ ) .
Explanation
Problem Analysis We are given the polynomial p ( x ) = 5 x 6 ( x + 2 ) 7 ( x − 3 ) 7 and the inequality x − 6 3 x + 12 ≥ 0 . We need to find the zeros and their multiplicities for the polynomial p ( x ) and solve the rational inequality.
Finding the Zeros of the Polynomial First, let's find the zeros of the polynomial p ( x ) . The zeros are the values of x for which p ( x ) = 0 . We have the factors 5 x 6 , ( x + 2 ) 7 , and ( x − 3 ) 7 . Setting each factor to zero gives us the zeros:
5 x 6 = 0 ⟹ x = 0 ( x + 2 ) 7 = 0 ⟹ x = − 2 ( x − 3 ) 7 = 0 ⟹ x = 3
Determining Multiplicities Now, let's determine the multiplicity of each zero. The multiplicity of a zero is the exponent of the corresponding factor.
For x = 0 , the factor is x 6 , so the multiplicity is 6. For x = − 2 , the factor is ( x + 2 ) 7 , so the multiplicity is 7. For x = 3 , the factor is ( x − 3 ) 7 , so the multiplicity is 7.
Finding Critical Points of the Inequality Next, let's solve the inequality x − 6 3 x + 12 ≥ 0 . First, we find the critical points by setting the numerator and denominator to zero:
3 x + 12 = 0 ⟹ 3 x = − 12 ⟹ x = − 4 x − 6 = 0 ⟹ x = 6
Creating a Sign Chart Now, we create a sign chart using the critical points x = − 4 and x = 6 . We consider the intervals ( − ∞ , − 4 ) , ( − 4 , 6 ) , and ( 6 , ∞ ) . We test a value in each interval to determine the sign of the expression x − 6 3 x + 12 in that interval.
Interval ( − ∞ , − 4 ) : Test x = − 5 . 0"> − 5 − 6 3 ( − 5 ) + 12 = − 11 − 15 + 12 = − 11 − 3 = 11 3 > 0 Interval ( − 4 , 6 ) : Test x = 0 . 0 − 6 3 ( 0 ) + 12 = − 6 12 = − 2 < 0 Interval ( 6 , ∞ ) : Test x = 7 . 0"> 7 − 6 3 ( 7 ) + 12 = 1 21 + 12 = 33 > 0
Determining the Solution Set The inequality is x − 6 3 x + 12 ≥ 0 . We want the intervals where the expression is positive or zero. The expression is positive in the intervals ( − ∞ , − 4 ) and ( 6 , ∞ ) . The expression is zero when x = − 4 . The expression is undefined when x = 6 , so we exclude x = 6 from the solution. Therefore, the solution to the inequality is ( − ∞ , − 4 ] ∪ ( 6 , ∞ ) .
Examples
Understanding the zeros and their multiplicities of a polynomial helps in analyzing the behavior of the polynomial function, such as where the graph crosses or touches the x-axis. Solving rational inequalities is useful in determining the domain of functions or finding intervals where a function is positive or negative. These concepts are applied in various fields like engineering, physics, and economics to model and analyze real-world phenomena.
The solution for the rational inequality x − 6 3 x + 12 ≥ 0 is ( − ∞ , − 4 ] ∪ ( 6 , ∞ ) . The polynomial p ( x ) = 5 x 6 ( x + 2 ) 7 ( x − 3 ) 7 has zeros x = 0 (multiplicity 6), x = − 2 (multiplicity 7), and x = 3 (multiplicity 7).
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