polynomial function in standard form with zeros calculator

The Rational Zero Theorem states that, if the polynomial $f(x)=a_nx^n+a_{n1}x^{n1}++a_1x+a_0$ has integer coefficients, then every rational zero of $f(x)$ has the form $\frac{p}{q}$ where $p$ is a factor of the constant term $a_0$ and $q$ is a factor of the leading coefficient $a_n$. The good candidates for solutions are factors of the last coefficient in the equation. Polynomial functions are expressions that may contain variables of varying degrees, coefficients, positive exponents, and constants. In this case we divide $ 2x^3 - x^2 - 3x - 6 $ by $ \color{red}{x - 2}$. Some examples of a linear polynomial function are f(x) = x + 3, f(x) = 25x + 4, and f(y) = 8y 3. , Find each zero by setting each factor equal to zero and solving the resulting equation. Lets walk through the proof of the theorem. There are various types of polynomial functions that are classified based on their degrees. Polynomial Factoring Calculator (shows all steps) supports polynomials with both single and multiple variables show help examples tutorial Enter polynomial: Examples: Check out all of our online calculators here! Example 3: Find a quadratic polynomial whose sum of zeros and product of zeros are respectively$\frac { 1 }{ 2 }$, 1 Sol. They are: Here is the polynomial function formula: f(x) = anxn + an-1xn-1 + + a2x2+ a1x + a0. Since f(x) = a constant here, it is a constant function. A polynomial is said to be in standard form when the terms in an expression are arranged from the highest degree to the lowest degree. Example 2: Find the zeros of f(x) = 4x - 8. Write the rest of the terms with lower exponents in descending order. WebThe calculator also gives the degree of the polynomial and the vector of degrees of monomials. The volume of a rectangular solid is given by $V=lwh$. Standard Form Polynomial 2 (7ab+3a^2b+cd^4) (2ef-4a^2)-7b^2ef Multivariate polynomial Monomial order Variables Calculation precision Exact Result Example: Put this in Standard Form: 3x 2 7 + 4x 3 + x 6. Let us look at the steps to writing the polynomials in standard form: Step 1: Write the terms. Learn how PLANETCALC and our partners collect and use data. The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. The highest degree of this polynomial is 8 and the corresponding term is 4v8. Write a polynomial function in standard form with zeros at 0,1, and 2? WebThus, the zeros of the function are at the point . Find zeros of the function: f x 3 x 2 7 x 20. The monomial x is greater than the x, since they are of the same degree, but the first is greater than the second lexicographically. If the degree is greater, then the monomial is also considered greater. What should the dimensions of the container be? Install calculator on your site. Standard form sorts the powers of #x# (or whatever variable you are using) in descending order. The polynomial can be written as, The quadratic is a perfect square. Although I can only afford the free version, I still find it worth to use. By definition, polynomials are algebraic expressions in which variables appear only in non-negative integer powers.In other words, the letters cannot be, e.g., under roots, in the denominator of a rational expression, or inside a function. Factor it and set each factor to zero. Write the rest of the terms with lower exponents in descending order. Example 1: A polynomial function of degree 5 has zeros of 2, -5, 1 and 3-4i.What is the missing zero? 2 x 2x 2 x; ( 3) Are zeros and roots the same? WebPolynomials Calculator. Webform a polynomial calculator First, we need to notice that the polynomial can be written as the difference of two perfect squares. By the Factor Theorem, the zeros of $x^36x^2x+30$ are 2, 3, and 5. Multiplicity: The number of times a factor is multiplied in the factored form of a polynomial. This is known as the Remainder Theorem. This pair of implications is the Factor Theorem. Linear Functions are polynomial functions of degree 1. There is a similar relationship between the number of sign changes in $f(x)$ and the number of negative real zeros. Both univariate and multivariate polynomials are accepted. The degree of a polynomial is the value of the largest exponent in the polynomial. Solve Now To solve a polynomial equation write it in standard form (variables and canstants on one side and zero on the other side of the equation). We find that algebraically by factoring quadratics into the form , and then setting equal to and , because in each of those cases and entire parenthetical term would equal 0, and anything times 0 equals 0. The standard form of a polynomial is expressed by writing the highest degree of terms first then the next degree and so on. However, it differs in the case of a single-variable polynomial and a multi-variable polynomial. Both univariate and multivariate polynomials are accepted. The graded reverse lexicographic order is similar to the previous one. example. If you are curious to know how to graph different types of functions then click here. Calculator shows detailed step-by-step explanation on how to solve the problem. WebHome > Algebra calculators > Zeros of a polynomial calculator Method and examples Method Zeros of a polynomial Polynomial = Solution Help Find zeros of a function 1. See, Polynomial equations model many real-world scenarios. Get step-by-step solutions from expert tutors as fast as 15-30 minutes. This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. By the Factor Theorem, we can write $f(x)$ as a product of $xc_1$ and a polynomial quotient. a n cant be equal to zero and is called the leading coefficient. We just need to take care of the exponents of variables to determine whether it is a polynomial function. WebPolynomial Standard Form Calculator The number 459,608 converted to standard form is 4.59608 x 10 5 Example: Convert 0.000380 to Standard Form Move the decimal 4 places to the right and remove leading zeros to get 3.80 a = Notice, written in this form, $xk$ is a factor of $f(x)$. However, #-2# has a multiplicity of #2#, which means that the factor that correlates to a zero of #-2# WebQuadratic function in standard form with zeros calculator The polynomial generator generates a polynomial from the roots introduced in the Roots field. \begin{aligned} x_1, x_2 &= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \\ x_1, x_2 &= \dfrac{-3 \pm \sqrt{3^2-4 \cdot 2 \cdot (-14)}}{2\cdot2} \\ x_1, x_2 &= \dfrac{-3 \pm \sqrt{9 + 4 \cdot 2 \cdot 14}}{4} \\ x_1, x_2 &= \dfrac{-3 \pm \sqrt{121}}{4} \\ x_1, x_2 &= \dfrac{-3 \pm 11}{4} \\ x_1 &= \dfrac{-3 + 11}{4} = \dfrac{8}{4} = 2 \\ x_2 &= \dfrac{-3 - 11}{4} = \dfrac{-14}{4} = -\dfrac{7}{2} \end{aligned} $$. According to the rule of thumbs: zero refers to a function (such as a polynomial), and the root refers to an equation. WebThis precalculus video tutorial provides a basic introduction into writing polynomial functions with given zeros. WebHow do you solve polynomials equations? This algebraic expression is called a polynomial function in variable x. The remainder is 25. This is a polynomial function of degree 4. WebPolynomial Factorization Calculator - Factor polynomials step-by-step. Exponents of variables should be non-negative and non-fractional numbers. Get detailed solutions to your math problems with our Polynomials step-by-step calculator. factor on the left side of the equation is equal to , the entire expression will be equal to . The real polynomial zeros calculator with steps finds the exact and real values of zeros and provides the sum and product of all roots. Multiply the linear factors to expand the polynomial. WebStandard form format is: a 10 b. For a polynomial, if #x=a# is a zero of the function, then # (x-a)# is a factor of the function. WebPolynomial Standard Form Calculator - Symbolab New Geometry Polynomial Standard Form Calculator Reorder the polynomial function in standard form step-by-step full pad 3x + x2 - 4 2. Precalculus Polynomial Functions of Higher Degree Zeros 1 Answer George C. Mar 6, 2016 The simplest such (non-zero) polynomial is: f (x) = x3 7x2 +7x + 15 Explanation: As a product of linear factors, we can define: f (x) = (x +1)(x 3)(x 5) = (x +1)(x2 8x + 15) = x3 7x2 +7x + 15 Roots =. Look at the graph of the function $f$ in Figure $\PageIndex{1}$. You can build a bright future by taking advantage of opportunities and planning for success. The number of positive real zeros is either equal to the number of sign changes of $f(x)$ or is less than the number of sign changes by an even integer. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. For example: 8x5 + 11x3 - 6x5 - 8x2 = 8x5 - 6x5 + 11x3 - 8x2 = 2x5 + 11x3 - 8x2. The coefficients of the resulting polynomial can be calculated in the field of rational or real numbers. Our online calculator, based on Wolfram Alpha system is able to find zeros of almost any, even very complicated function. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. Sol. The first monomial x is lexicographically greater than second one x, since after subtraction of exponent tuples we obtain (0,1,-2), where leftmost nonzero coordinate is positive. The like terms are grouped, added, or subtracted and rearranged with the exponents of the terms in descending order. You can also verify the details by this free zeros of polynomial functions calculator. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Third Edition, 2007, Springer, Everyone who receives the link will be able to view this calculation, Copyright PlanetCalc Version: Look at the graph of the function $f$ in Figure $\PageIndex{2}$. Again, there are two sign changes, so there are either 2 or 0 negative real roots. Arranging the exponents in descending order, we get the standard polynomial as 4v8 + 8v5 - v3 + 8v2. All the roots lie in the complex plane. The Rational Zero Theorem tells us that if $\frac{p}{q}$ is a zero of $f(x)$, then $p$ is a factor of 3 and $q$ is a factor of 3. This is a polynomial function of degree 4. The degree of the polynomial function is determined by the highest power of the variable it is raised to. Please enter one to five zeros separated by space. We solved each of these by first factoring the polynomial and then using the zero factor property on the factored form. Double-check your equation in the displayed area. Radical equation? Check. So to find the zeros of a polynomial function f(x): Consider a linear polynomial function f(x) = 16x - 4. Use the Rational Zero Theorem to list all possible rational zeros of the function. You don't have to use Standard Form, but it helps. Rational equation? The standard form of polynomial is given by, f(x) = anxn + an-1xn-1 + an-2xn-2 + + a1x + a0, where x is the variable and ai are coefficients. \[f(\dfrac{1}{2})=2{(\dfrac{1}{2})}^3+{(\dfrac{1}{2})}^24(\dfrac{1}{2})+1=3\]. 4)it also provide solutions step by step. How to: Given a polynomial function $f$, use synthetic division to find its zeros. The cake is in the shape of a rectangular solid. 3x + x2 - 4 2. This is true because any factor other than $x(abi)$, when multiplied by $x(a+bi)$, will leave imaginary components in the product. Further polynomials with the same zeros can be found by multiplying the simplest polynomial with a factor. Example 02: Solve the equation $ 2x^2 + 3x = 0 $. To write a polynomial in a standard form, the degree of the polynomial is important as in the standard form of a polynomial, the terms are written in decreasing order of the power of x. In this case we have $ a = 2, b = 3 , c = -14 $, so the roots are: $$ Has helped me understand and be able to do my homework I recommend everyone to use this. Write the term with the highest exponent first. Definition of zeros: If x = zero value, the polynomial becomes zero. se the Remainder Theorem to evaluate $f(x)=2x^53x^49x^3+8x^2+2$ at $x=3$. According to the rule of thumbs: zero refers to a function (such as a polynomial), and the root refers to an equation. Example 04: Solve the equation $ 2x^3 - 4x^2 - 3x + 6 = 0 $. WebPolynomials Calculator. Learn the why behind math with our certified experts, Each exponent of variable in polynomial function should be a. If the remainder is 0, the candidate is a zero. Example 2: Find the degree of the monomial: - 4t. The graph shows that there are 2 positive real zeros and 0 negative real zeros. If k is a zero, then the remainder r is f(k) = 0 and f(x) = (x k)q(x) + 0 or f(x) = (x k)q(x). The process of finding polynomial roots depends on its degree. Reset to use again. The bakery wants the volume of a small cake to be 351 cubic inches. There will be four of them and each one will yield a factor of $f(x)$. Write the polynomial as the product of $(xk)$ and the quadratic quotient. The monomial x is greater than x, since the degree ||=7 is greater than the degree ||=6. Rational root test: example. WebPolynomial Factorization Calculator - Factor polynomials step-by-step. See, According to the Rational Zero Theorem, each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. This free math tool finds the roots (zeros) of a given polynomial. Each equation type has its standard form. A polynomial degree deg(f) is the maximum of monomial degree || with nonzero coefficients. Function's variable: Examples. In this article, we will learn how to write the standard form of a polynomial with steps and various forms of polynomials. The first one is obvious. In the last section, we learned how to divide polynomials. The zero at #x=4# continues through the #x#-axis, as is the case In this case, $f(x)$ has 3 sign changes. This is the standard form of a quadratic equation, $$ x_1, x_2 = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} $$, Example 01: Solve the equation $ 2x^2 + 3x - 14 = 0 $. WebZeros: Values which can replace x in a function to return a y-value of 0. Then, by the Factor Theorem, $x(a+bi)$ is a factor of $f(x)$. Write a polynomial function in standard form with zeros at 0,1, and 2? The polynomial can be up to fifth degree, so have five zeros at maximum. WebZero: A zero of a polynomial is an x-value for which the polynomial equals zero. Either way, our result is correct. most interesting ones are: quadratic - degree 2, Cubic - degree 3, and Quartic - degree 4. In this article, let's learn about the definition of polynomial functions, their types, and graphs with solved examples. Examples of graded reverse lexicographic comparison: Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The Factor Theorem is another theorem that helps us analyze polynomial equations. For a function to be a polynomial function, the exponents of the variables should neither be fractions nor be negative numbers. The polynomial can be up to fifth degree, so have five zeros at maximum. It tells us how the zeros of a polynomial are related to the factors. Check. How do you find the multiplicity and zeros of a polynomial? In a multi-variable polynomial, the degree of a polynomial is the highest sum of the powers of a term in the polynomial. Example 1: A polynomial function of degree 5 has zeros of 2, -5, 1 and 3-4i.What is the missing zero? WebThis calculator finds the zeros of any polynomial. Note that if f (x) has a zero at x = 0. then f (0) = 0. What is polynomial equation? $$ \begin{aligned} 2x^2 + 3x &= 0 \\ \color{red}{x} \cdot \left( \color{blue}{2x + 3} \right) &= 0 \\ \color{red}{x = 0} \,\,\, \color{blue}{2x + 3} & \color{blue}{= 0} \\ WebFind the zeros of the following polynomial function: \[ f(x) = x^4 4x^2 + 8x + 35 \] Use the calculator to find the roots. Graded lex order examples: Example 3: Find the degree of the polynomial function f(y) = 16y5 + 5y4 2y7 + y2. Standard form sorts the powers of #x# (or whatever variable you are using) in descending order. If $2+3i$ were given as a zero of a polynomial with real coefficients, would $23i$ also need to be a zero? The zeros are $4$, $\frac{1}{2}$, and $1$. However, with a little bit of practice, anyone can learn to solve them. Practice your math skills and learn step by step with our math solver. Sum of the zeros = 4 + 6 = 10 Product of the zeros = 4 6 = 24 Hence the polynomial formed = x 2 (sum of zeros) x + Product of zeros = x 2 10x + 24 Let the cubic polynomial be ax3 + bx2 + cx + d x3+ $\frac { b }{ a }$x2+ $\frac { c }{ a }$x + $\frac { d }{ a }$(1) and its zeroes are , and then + + = 2 =$\frac { -b }{ a }$ + + = 7 = $\frac { c }{ a }$ = 14 =$\frac { -d }{ a }$ Putting the values of $\frac { b }{ a }$, $\frac { c }{ a }$, and $\frac { d }{ a }$ in (1), we get x3+ (2) x2+ (7)x + 14 x3 2x2 7x + 14, Example 7: Find the cubic polynomial with the sum, sum of the product of its zeroes taken two at a time and product of its zeroes as 0, 7 and 6 respectively. Form A Polynomial With The Given Zeros Example Problems With Solutions Example 1: Form the quadratic polynomial whose zeros are 4 and 6. Click Calculate. Input the roots here, separated by comma. A new bakery offers decorated sheet cakes for childrens birthday parties and other special occasions. With Cuemath, you will learn visually and be surprised by the outcomes. Double-check your equation in the displayed area. Calculus: Fundamental Theorem of Calculus, Factoring-polynomials.com makes available insightful info on standard form calculator, logarithmic functions and trinomials and other algebra topics. Let us set each factor equal to 0, and then construct the original quadratic function absent its stretching factor. The terms have variables, constants, and exponents. We've already determined that its possible rational roots are 1/2, 1, 2, 3, 3/2, 6. To find its zeros: Consider a quadratic polynomial function f(x) = x2 + 2x - 5. Given the zeros of a polynomial function $f$ and a point $(c, f(c))$ on the graph of $f$, use the Linear Factorization Theorem to find the polynomial function. Calculator shows detailed step-by-step explanation on how to solve the problem. The polynomial can be up to fifth degree, so have five zeros at maximum. Roots calculator that shows steps. Hence the degree of this particular polynomial is 4. For example, the degree of polynomial $ p(x) = 8x^\color{red}{2} + 3x -1 $ is $\color{red}{2}$. solution is all the values that make true. Determine math problem To determine what the math problem is, you will need to look at the given Use the Rational Zero Theorem to list all possible rational zeros of the function. Here, a n, a n-1, a 0 are real number constants. Find zeros of the function: f x 3 x 2 7 x 20. The degree of the polynomial function is determined by the highest power of the variable it is raised to. However, #-2# has a multiplicity of #2#, which means that the factor that correlates to a zero of #-2# is represented in the polynomial twice. The solutions are the solutions of the polynomial equation. 2 x 2x 2 x; ( 3) WebZeros: Values which can replace x in a function to return a y-value of 0. This tells us that $k$ is a zero. Finding the zeros of cubic polynomials is same as that of quadratic equations. Since $xc_1$ is linear, the polynomial quotient will be of degree three. ( 6x 5) ( 2x + 3) Go! Find the zeros of the quadratic function. It is written in the form: ax^2 + bx + c = 0 where x is the variable, and a, b, and c are constants, a 0. Webwrite a polynomial function in standard form with zeros at 5, -4 . 1 Answer Douglas K. Apr 26, 2018 #y = x^3-3x^2+2x# Explanation: If #0, 1, and 2# are zeros then the following is factored form: #y = (x-0)(x-1)(x-2)# Multiply: #y = (x)(x^2-3x+2)# #y = x^3-3x^2+2x# Answer link. We can use synthetic division to test these possible zeros. The possible values for $\dfrac{p}{q}$, and therefore the possible rational zeros for the function, are 3,1, and $\dfrac{1}{3}$. According to the Linear Factorization Theorem, a polynomial function will have the same number of factors as its degree, and each factor will be in the form $(xc)$, where $c$ is a complex number. a is a number whose absolute value is a decimal number greater than or equal to 1, and less than 10: 1 | a | < 10. b is an integer and is the power of 10 required so that the product of the multiplication in standard form equals the original number. Function's variable: Examples. Write the term with the highest exponent first. If you plug in -6, 2, or 5 to x, this polynomial you are trying to find becomes zero. Lets begin by testing values that make the most sense as dimensions for a small sheet cake. We have two unique zeros: #-2# and #4#. Addition and subtraction of polynomials are two basic operations that we use to increase or decrease the value of polynomials. For example: 14 x4 - 5x3 - 11x2 - 11x + 8. And if I don't know how to do it and need help. The four most common types of polynomials that are used in precalculus and algebra are zero polynomial function, linear polynomial function, quadratic polynomial function, and cubic polynomial function. While a Trinomial is a type of polynomial that has three terms. What is the value of x in the equation below? The types of polynomial terms are: Constant terms: terms with no variables and a numerical coefficient. Radical equation? Group all the like terms. WebHow do you solve polynomials equations? By the Factor Theorem, these zeros have factors associated with them. WebFree polynomal functions calculator The number 459,608 converted to standard form is 4.59608 x 10 5 Example: Convert 0.000380 to Standard Form Move the decimal 4 places to the right and remove leading zeros to get 3.80 a = What our students say John Tillotson Best calculator out there. d) f(x) = x2 - 4x + 7 = x2 - 4x1/2 + 7 is NOT a polynomial function as it has a fractional exponent for x. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. Let the cubic polynomial be ax3 + bx2 + cx + d x3+ $\frac { b }{ a }$x2+ $\frac { c }{ a }$x + $\frac { d }{ a }$(1) and its zeroes are , and then + + = 0 =$\frac { -b }{ a }$ + + = 7 = $\frac { c }{ a }$ = 6 =$\frac { -d }{ a }$ Putting the values of $\frac { b }{ a }$, $\frac { c }{ a }$, and $\frac { d }{ a }$ in (1), we get x3 (0) x2+ (7)x + (6) x3 7x + 6, Example 8: If and are the zeroes of the polynomials ax2 + bx + c then form the polynomial whose zeroes are $\frac { 1 }{ \alpha } \quad and\quad \frac { 1 }{ \beta }$ Since and are the zeroes of ax2 + bx + c So + = $\frac { -b }{ a }$, = $\frac { c }{ a }$ Sum of the zeroes = $\frac { 1 }{ \alpha } +\frac { 1 }{ \beta } =\frac { \alpha +\beta }{ \alpha \beta } $ $=\frac{\frac{-b}{c}}{\frac{c}{a}}=\frac{-b}{c}$ Product of the zeroes $=\frac{1}{\alpha }.\frac{1}{\beta }=\frac{1}{\frac{c}{a}}=\frac{a}{c}$ But required polynomial is x2 (sum of zeroes) x + Product of zeroes $\Rightarrow {{\text{x}}^{2}}-\left( \frac{-b}{c} \right)\text{x}+\left( \frac{a}{c} \right)$ $\Rightarrow {{\text{x}}^{2}}+\frac{b}{c}\text{x}+\frac{a}{c}$ $\Rightarrow c\left( {{\text{x}}^{2}}+\frac{b}{c}\text{x}+\frac{a}{c} \right)$ cx2 + bx + a, Filed Under: Mathematics Tagged With: Polynomials, Polynomials Examples, ICSE Previous Year Question Papers Class 10, ICSE Specimen Paper 2021-2022 Class 10 Solved, Concise Mathematics Class 10 ICSE Solutions, Concise Chemistry Class 10 ICSE Solutions, Concise Mathematics Class 9 ICSE Solutions, Class 11 Hindi Antra Chapter 9 Summary Bharatvarsh Ki Unnati Kaise Ho Sakti Hai Summary Vyakhya, Class 11 Hindi Antra Chapter 8 Summary Uski Maa Summary Vyakhya, Class 11 Hindi Antra Chapter 6 Summary Khanabadosh Summary Vyakhya, John Locke Essay Competition | Essay Competition Of John Locke For Talented Ones, Sangya in Hindi , , My Dream Essay | Essay on My Dreams for Students and Children, Viram Chinh ( ) in Hindi , , , EnvironmentEssay | Essay on Environmentfor Children and Students in English. Answer: 5x3y5+ x4y2 + 10x in the standard form. \[ \begin{align*} \dfrac{p}{q}=\dfrac{factor\space of\space constant\space term}{factor\space of\space leading\space coefficient} \\[4pt] &=\dfrac{factor\space of\space 3}{factor\space of\space 3} \end{align*}\]. This tells us that $f(x)$ could have 3 or 1 negative real zeros. Each equation type has its standard form. Let $f$ be a polynomial function with real coefficients, and suppose $a +bi$, $b0$, is a zero of $f(x)$. WebIn math, a quadratic equation is a second-order polynomial equation in a single variable. Install calculator on your site. The calculator computes exact solutions for quadratic, cubic, and quartic equations. 2 x 2x 2 x; ( 3) Find the exponent. b) Determine all possible values of $\dfrac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient. Write the polynomial as the product of factors. This algebraic expression is called a polynomial function in variable x. The standard form of a polynomial is given by, f(x) = anxn + an-1xn-1 + an-2xn-2 + + a1x + a0. WebPolynomials involve only the operations of addition, subtraction, and multiplication. ( 6x 5) ( 2x + 3) Go! Here the polynomial's highest degree is 5 and that becomes the exponent with the first term. Example: Put this in Standard Form: 3x 2 7 + 4x 3 + x 6. Factor it and set each factor to zero. The below-given image shows the graphs of different polynomial functions. Function zeros calculator. Let's see some polynomial function examples to get a grip on what we're talking about:. If k is a zero, then the remainder r is f(k) = 0 and f(x) = (x k)q(x) + 0 or f(x) = (x k)q(x). Any polynomial in #x# with these zeros will be a multiple (scalar or polynomial) of this #f(x)# . A polynomial function in standard form is: f(x) = anxn + an-1xn-1 + + a2x2+ a1x + a0. WebFactoring-polynomials.com makes available insightful info on standard form calculator, logarithmic functions and trinomials and other algebra topics. And, if we evaluate this for $x=k$, we have, \[\begin{align*} f(k)&=(kk)q(k)+r \\[4pt] &=0{\cdot}q(k)+r \\[4pt] &=r \end{align*}\].