College Algebra – Review 3 Polynomial Equation Name: ________________________________________ _____ 1. Find the domain of a function. Express as an interval. f(𝑥)=2𝑥−3 √3−12𝑥 Given the function g(x) = −6𝑥4+5𝑥2+𝑥+32, determine the leading term, degree, leading coefficient, and end behavior of the function. The leading term : Degree: The leading coefficient : End Behavior: College Algebra – Review 3 2. Divide 18𝑥4+9𝑥3+3𝑥2 by 3𝑥2+1 using long division . Quotient: Remainder: Use synthetic division to divide . (𝑥2−5𝑥−5𝑥3+𝑥4) ÷( 5+𝑥) Quotient: Remainder: College Algebra – Review 3 3. Use the Rational Zero Theorem to list all possible rational zeros for f(x) = 3𝑥5−11𝑥4−6𝑥+12 Use the Remainder Theorem to find the indicated function value. f(x) = 4𝑥3−2𝑥2−6𝑥+3 ; f(−1) College Algebra – Review 3 4. Find the zeros of f(x) = −2(𝑥−5)2(𝑥+7)3 and give the multiplicity of each zero. State whether the graph crosses the x -axis or touches the x-axis and turns around at each zero. Zero Its multiplicity Behavior Use the table below to find: a) (f∘g)(7) = b) (g∘f)(0) = c) (f∘f)(−3) = d) (g∘g)(5) = x -3 0 1 3 5 7 9 11 f(x) 0 7 5 3 11 -3 1 9 g(x) 3 -3 1 9 11 1 5 0 College Algebra – Review 3 5. Given the functions: f(x) = −6x g(x) = x − 5 h(x) = −3𝑥2+ 9x + 30. Determine each of the following. Give your answers as simplified expressions written in descending order. a) Find and simplify g(x) + h(x) b) Find and simplify h(x) − g(x) c) Find and simplify f(x) ⋅ h(x) d) Find and simplify h(x) g(x) e) The domain restriction for f(x) g(x) Use interval notation to express the solution. College Algebra – Review 3 6. Given 𝑓(𝑥)=1 𝑥+3 and 𝑔(𝑥)=2 𝑥−3, find each of the following: a) (𝑓∘𝑔)(𝑥) b) (𝑔∘𝑓)(𝑥) College Algebra – Review 3 7. Find all zeros of f(x) =6𝑥3−11𝑥2+6𝑥−1. a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part b to find the remaining zeros of the polynomial function. d) List all zeros. College Algebra – Review 3 8. Solve 𝑥4−2𝑥2−16𝑥−15=0. a) List all possible rational roots . b) Use synthetic division to test the possible rational roots and find an actual roots . c) Use the quotient from part b to find the remaining roots and solve the equation . d) List all zeros. College Algebra – Review 3 9. Sketch a possible graph for 𝑓(𝑥)= 1 4(𝑥−2)(𝑥+3)2 a) 𝑦 -intercept: b) 𝑥 -intercepts: c) Use the multiplicities of the zeros to determine the behavior of the polynomial at the x -intercepts. d) Determine the end behavior by examining the leading term. The leading coefficient is ____________ As x → −∞ , f(X) → _________ The degree of the polynomial _________ As x → ∞ , f(X) → _________ e) Find the domain. f) Sketch the graph. College Algebra – Review 3 10. Solve 𝑥3−2𝑥2−15𝑥+36 > 0 algebraically , graph and write the solution in interval notation. College Algebra – Review 3 11. Solve −𝑥2+6𝑥 ≤ 8 graphically and write the solution in interval notation.