Rational exponents and radicals #4 - Questions and Answers

Question 1 (a) find the rational zeros and then the other zeros of the polynomial function f(x) = x³ - 4x² + 2x + 4, that is, solve f(x) = 0. (type an exact answer, using radicals as needed. Simplify your answer. Use a comma to separate answers as needed.) (b) Factor f(x) into linear factors. (type an exact answer, using radicals as needed. Simplify your answer. Type an expression using x as the variable.) Answer Use graph f(x) = x³-4x²+2x+4 using a graphing calculator to find the rational zeros It cuts the graph at x = 2 Means x = 2 is a zero Then we use x=2 to find the other factor. We use synthetic division. 1 -4 2 4 2 2 -4 -4 1 -2 -2 | 0 Another factor is x²-2x-2 Then we use the quadratic formula to find the zeros from it x²-2x-2=0 x = (2 ± √(-2)² - 4(1)(-2)) / 2(1) x = (2 ± √12) / 2 x = (2 ± 2√3) / 2 x = 1 ± √3 Answer(a): 1-√3, 2, 1+√3 b) Linear factored form of f is: f(x) = [x-(1-√3)] (x-2) [x-(1+√3)] f(x) = (x-1+√3) (x-2)(x-1-√3) Question 2 Use rational exponents to simplify. Write the answer in radical notation if appropriate (√cd)¹⁴ Answer Given: (√cd)¹⁴ We need use property of rational exponents (√a)ᵇ = (a^(1/2))ᵇ = a^( b/2 ) use property of rational exponents (√a)ᵇ = (a^(1/2))ᵇ = a^( b/2 ) (√cd)¹⁴ = (cd)^(14/2) = (cd)⁷ = c⁷d⁷ Finally answer is: (cd)⁷ = c⁷d⁷ Question 3 To calculate: The restriction on the variable for rational expression 6c / (7a³b²) Answer The restriction on the variable for rational expression 6c / (7a³b²) are a ≠ 0 and b ≠ 0. Given Information: The provided rational expression is 6c / (7a³b²) Formula used: The standard form of rational expression is p(x) / q(x) Where, p (x) and q(x) are polynomial and q(x) ≠ 0. Calculation: Consider the rational expression, 6c / (7a³b²). For the expression 6c / (7a³b²), if a = 0 were substituted for a, the denominator would be zero. If b = 0 were substituted for b, the denominator would be zero. Hence, for this expression 6c / (7a³b²), a ≠ 0 and b≠ 0 If, either a or b were zero. Then the product would be zero. Therefore, the restriction on the variable for rational expression 6c / (7a³b²), are a ≠ 0 and b ≠ 0. Question 4 Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. Find the real zeros of f(x) = 5x³- x² + 5x - 1 Answer Given: find all real zeros of f(x) = 5x³ - x² + 5x - 1, and use the zeros to factor f over the real numbers. Used concept: Rational zeros theorem: Let P(x) is a polynomial. if p/q is a zero of P(x) then p is the factor of the constant term of P(x) and q is the factor of the leading coefficient of P(x). Let p/q is the rational zero of the given polynomial. Here, p is a factor of the constant term -1 and q is a factor of the leading coefficient 5. So, the possible value of p = ±1 and q = ± 1, ± 5 Now, the possible value of p/q = ± 1, ± 1/5 Now, check the value of function f(x) = 5x³ - x² + 5x - 1 f(1) = 5(1)³ - (1)² + 5(1) - 1 = 8 ≠ 0 f(-1) = 5(-1)³ - (-1)² + 5(-1) - 1 = - 12 ≠ 0 f(1/5) = 5(1/5)³ - (1/5)² + 5(1/5) - 1 = 0 f(-1/5) = 5(-1/5)³ - (-1/5)² + 5(-1/5) - 1 = - 26/25 ≠ 0 So, the zero of the given function is 1/5 in real numbers. f(x) = 5x³ - x² + 5x - 1 f(x) = x²(5x - 1) + 1(5x - 1) f(x) = (5x - 1)(x² + 1) Finally answer is: (5x - 1)(x² + 1) Question 5 Find the exact value of each of the remaining trigonometric function of θ. cos θ = 24 / 25 , 270° < θ < 360° sin θ =? tan θ =? sec θ =? csc θ =? cot θ =? Answer Step 1 Given: cos θ = 24/25, 270° <θ < 360° So the angle lies in fourth quadrant. In the fourth quadrant cosine and secant are positive all others are negative. Pythagorean identity is used for finding sine value sin²θ + cos²θ = 1 sin θ = ±√1 - cos²θ = ±√1 - (24/25)² = ±√(25² - 24²) / 25² = ±√49 / 25 = ±7 / 25 The sine value is negative since angle is in fourth quadrant. Step 2 The tan value can be found as tan θ = sin θ / cos θ = - 7/25 / 24/25 = -7 / 24 Step 3 Secant is reciprocal of cosine. sec θ = 1 / cos θ = 1 / (24/25) = 25 / 24 Step 4 Cosecant is the reciprocal of sine. csc θ = 1 / sin θ = 1 / (-7/25) = -25 / 7 Step 5 Cot is the reciprocal of tan cot θ = 1 / tan θ = 1 / (-7/24) = -24 / 7