Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?
State the possible rational roots for each function
\(\textbf{1)}\) \(f(x)=x^5+3x^3+2x^2+5x-12\) 
\(\textbf{2)}\) \(f(x)=4x^5+3x^3+2x^2+5x-12\)
\(\textbf{3)}\) \(f(x)=7x^8+9x^4+3x^3-2x+4\)
\(\textbf{4)}\) \(f(x)=x^3+9x^2+2x+16\)
\(\textbf{5)}\) \(f(x)=-8x^4+-2x^2+x-1\)
Find all zeroes of each function
\(\textbf{6)}\) \(f(x)=x^3+2x^2-11x-12\)
\(\textbf{7)}\) \(f(x)=x^3-6x^2+11x-6\)
\(\textbf{8)}\) \(f(x)=2x^3+x^2-5x+2\)
Express each in linear factored form
\(\textbf{9)}\) Express \(f(x)=x^3+2x^2-11x-12\) in linear factored form.
\(\textbf{10)}\) Express \(f(x)=x^3-6x^2+11x-6\) in linear factored form.
\(\textbf{11)}\) Express \(f(x)=2x^3+x^2-5x+2\) in linear factored form.
See Related Pages\(\)
\(\bullet\text{ Adding and Subtracting Polynomials}\)
\(\,\,\,\,\,\,\,\,(4d+7)−(2d−5)…\)
\(\bullet\text{ Multiplying Polynomials}\)
\(\,\,\,\,\,\,\,\,(x+2)(x^2+3x−5)…\)
\(\bullet\text{ Dividing Polynomials}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)
\(\bullet\text{ Dividing Polynomials (Synthetic Division)}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)
\(\bullet\text{ Synthetic Substitution}\)
\(\,\,\,\,\,\,\,\,f(x)=4x^4−3x^2+8x−2…\)
\(\bullet\text{ End Behavior}\)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow \infty, \quad f(x)\rightarrow \infty \)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow -\infty, \quad f(x)\rightarrow \infty… \)
\(\bullet\text{ Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+10x−24=0…\)
\(\bullet\text{ Quadratic Formula and the Discriminant}\)
\(\,\,\,\,\,\,\,\,x=-b \pm \displaystyle\frac{\sqrt{b^2-4ac}}{2a}…\)
\(\bullet\text{ Complex Numbers}\)
\(\,\,\,\,\,\,\,\,i=\sqrt{-1}…\)
\(\bullet\text{ Multiplicity of Roots}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Rational Zero Theorem}\)
\(\,\,\,\,\,\,\,\, \pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12…\)
\(\bullet\text{ Descartes Rule of Signs}\)
\(\,\)
\(\bullet\text{ Roots and Zeroes}\)
\(\,\,\,\,\,\,\,\,\text{Solve for }x. 3x^2+4x=0…\)
\(\bullet\text{ Linear Factored Form}\)
\(\,\,\,\,\,\,\,\,f(x)=(x+4)(x+1)(x−3)…\)
\(\bullet\text{ Polynomial Inequalities}\)
\(\,\,\,\,\,\,\,\,x^3-4x^2-4x+16 \gt 0…\)
In Summary
The Rational Zero Theorem states that if a polynomial (with rational coefficients) has a rational zero, then that zero can be expressed in the form of a fraction p/q, where p is a factor of the constant term of the polynomial and q is a factor of the leading coefficient of the polynomial. It is usually used when the degree of a polynomial is greater than 2 and traditional methods of factoring won’t work. It is commonly used in conjunction with polynomial division or synthetic division.
The Rational Zero Theorem is typically introduced in an algebra 2 or pre-calculus course along with the chapter on polynomials.
About Andymath.com
Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.
Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.
Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!
Thank you for visiting. How exciting!