Add two to both sides, Now divide factors of the leadings with factors of the constant. Ic an tell you a way that works for it though, in fact my prefered way works for all quadratics, and that i why it is my preferred way. Well leave it to our readers to check these results. then volume of, A: Triangle law of cosine We will now explore how we can find the zeros of a polynomial by factoring, followed by the application of the zero product property. to factor this expression right over here, this As you can see in Figure \(\PageIndex{1}\), the graph of the polynomial crosses the horizontal axis at x = 6, x = 1, and x = 5. The real polynomial zeros calculator with steps finds the exact and real values of zeros and provides the sum and product of all roots. Show your work. Since the function equals zero when is , one of the factors of the polynomial is . Let's look at a more extensive example. However, note that knowledge of the end-behavior and the zeros of the polynomial allows us to construct a reasonable facsimile of the actual graph. whereS'x is the rate of annual saving andC'x is the rate of annual cost. (Enter your answers as a comma-separated list. I hope this helps. For example, suppose we have a polynomial equation. F12 m(x) =x35x2+ 12x+18 If there is more than one answer, separate them with commas. Let us find the quotient on dividing x3 + 13 x2 + 32 x + 20 by ( x + 1). H Direct link to XGR (offline)'s post There might be other ways, Posted 2 months ago. Factorise : x3+13x2+32x+20 3.1. So this is going to be five x times, if we take a five x out F4 F We say that \(a\) is a zero of the polynomial if and only if \(p(a) = 0\). Prt S A polynomial with rational coefficients can sometimes be written as a product of lower-degree polynomials that also have rational coefficients. But the key here is, lets $\exponential{(x)}{3} + 13 \exponential{(x)}{2} + 32 x + 20 $. @ From the source of Wikipedia: Zero of a function, Polynomial roots, Fundamental theorem of algebra, Zero set. A special multiplication pattern that appears frequently in this text is called the difference of two squares. (Remember that this is . Lets begin with a formal definition of the zeros of a polynomial. Write the resulting polynomial in standard form and . How did we get (x+3)(x-2) from (x^2+x-6)? Here is an example of a 3rd degree polynomial we can factor by first taking a common factor and then using the sum-product pattern. In this section we concentrate on finding the zeros of the polynomial. Factoring Calculator. Identify the Zeros and Their Multiplicities h(x)=2x^4-13x^3+32x^2-53x+20 G Step 2. Well if we divide five, if Lets look at a final example that requires factoring out a greatest common factor followed by the ac-test. It means (x+2) is a factor of given polynomial. The zeros of a polynomial calculator can find all zeros or solution of the polynomial equation P (x) = 0 by setting each factor to 0 and solving for x. Wolfram|Alpha is a great tool for factoring, expanding or simplifying polynomials. Well find the Difference of Squares pattern handy in what follows. Q \[\begin{aligned}(a+b)(a-b) &=a(a-b)+b(a-b) \\ &=a^{2}-a b+b a-b^{2} \end{aligned}\]. Enter the expression you want to factor in the editor. b) Use synthetic division or the remainder theorem to show that is a factor of /(r) c) Find the remaining zeros. You should always look to factor out the greatest common factor in your first step. C Login. Lets examine the connection between the zeros of the polynomial and the x-intercepts of the graph of the polynomial. However, the original factored form provides quicker access to the zeros of this polynomial. Factories: x 3 + 13 x 2 + 32 x + 20. times this second degree, the second degree expression If we take out a five x If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Thats why we havent scaled the vertical axis, because without the aid of a calculator, its hard to determine the precise location of the turning points shown in Figure \(\PageIndex{2}\). { "6.01:_Polynomial_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "6.02:_Zeros_of_Polynomials" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "6.03:_Extrema_and_Models" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Preliminaries" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Linear_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Absolute_Value_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Quadratic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Polynomial_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Rational_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Exponential_and_Logarithmic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Radical_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "x-intercept", "license:ccbyncsa", "showtoc:no", "roots", "authorname:darnold", "zero of the polynomial", "licenseversion:25" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FAlgebra%2FIntermediate_Algebra_(Arnold)%2F06%253A_Polynomial_Functions%2F6.02%253A_Zeros_of_Polynomials, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), The x-intercepts and the Zeros of a Polynomial, status page at https://status.libretexts.org, x 3 is a factor, so x = 3 is a zero, and. A Start your trial now! In this problem that common factor is 5, so we can factor it out to get 5(x - x - 6). (x2 - (5)^2) is . Then we can factor again to get 5((x - 3)(x + 2)). 3x3+x2-3x-12. It explains how to find all the zeros of a polynomial function. View this solution and millions of others when you join today! And if we take out a P (x) = 2.) stly cloudy How to find all the zeros of polynomials? Again, note how we take the square root of each term, form two binomials with the results, then separate one pair with a plus, the other with a minus. Direct link to NEOVISION's post p(x)=2x^(3)-x^(2)-8x+4 Since we obtained x+1as one of the factors, we should regroup the terms of given polynomial accordingly. There are three solutions: x_0 = 2 x_1 = 3+2i x_2 = 3-2i The rational root theorem tells us that rational roots to a polynomial equation with integer coefficients can be written in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Thus, either, \[x=-3 \quad \text { or } \quad x=2 \quad \text { or } \quad x=5\]. Further, Hence, the factorization of . We have no choice but to sketch a graph similar to that in Figure \(\PageIndex{4}\). Polynomials with rational coefficients always have as many roots, in the complex plane, as their degree; however, these roots are often not rational numbers. Again, it is very important to realize that once the linear (first degree) factors are determined, the zeros of the polynomial follow. You might ask how we knew where to put these turning points of the polynomial. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. +1, + For each of the polynomials in Exercises 35-46, perform each of the following tasks. This doesn't help us find the other factors, however. Lets use these ideas to plot the graphs of several polynomials. Solve for . the interactive graph. Thus, the zeros of the polynomial are 0, 3, and 5/2. = x 3 + 13x 2 + 32x + 20 Put x = -1 in p(x), we get p(-1) = (-1) 3 + 13(-1) 2 + 32(-1) + 20 Login. If you're seeing this message, it means we're having trouble loading external resources on our website. The graph must therefore be similar to that shown in Figure \(\PageIndex{6}\). The polynomial is not yet fully factored as it is not yet a product of two or more factors. Consequently, the zeros of the polynomial are 0, 4, 4, and 2. More than just an online factoring calculator. The first factor is the difference of two squares and can be factored further. A: Here the total tuition fees is 120448. Could you also factor 5x(x^2 + x - 6) as 5x(x+2)(x-3) = 0 to get x=0, x= -2, and x=3 instead of factoring it as 5x(x+3)(x-2)=0 to get x=0, x= -3, and x=2? Q. x3 + 13x2 + 32x + 20. F1 Z DelcieRiveria Answer: The all zeroes of the polynomial are -10, -2 and -1. Find the zeros of the polynomial \[p(x)=4 x^{3}-2 x^{2}-30 x\]. To view the purposes they believe they have legitimate interest for, or to object to this data processing use the vendor list link below. Once youve mastered multiplication using the Difference of Squares pattern, it is easy to factor using the same pattern. Identify the Zeros and Their Multiplicities x^3-6x^2+13x-20. Step 1: Find a factor of the given polynomial. about what the graph could be. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. 009456 Find all the zeros. And let's see, positive In Exercises 7-28, identify all of the zeros of the given polynomial without the aid of a calculator. makes five x equal zero. A: Let three sides of the parallelepiped are denoted by vectors a,b,c Once you've done that, refresh this page to start using Wolfram|Alpha. Standard IX Mathematics. The graph and window settings used are shown in Figure \(\PageIndex{7}\). B Step 1.5. Divide by . The converse is also true, but we will not need it in this course. And it is the case. The polynomial \(p(x)=x^{4}+2 x^{3}-16 x^{2}-32 x\) has leading term \(x^4\). Find all the zeros of the polynomial x^3 + 13x^2 +32x +20. brainly.in/question/27985 Advertisement abhisolanki009 Answer: hey, here is your solution. We have one at x equals, at x equals two. A: cos=-3989isinthethirdquadrant QnA. Note that there are two turning points of the polynomial in Figure \(\PageIndex{2}\). Note that this last result is the difference of two terms. For now, lets continue to focus on the end-behavior and the zeros. Identify the Conic 25x^2+9y^2-50x-54y=119, Identify the Zeros and Their Multiplicities x^4+7x^3-22x^2+56x-240, Identify the Zeros and Their Multiplicities d(x)=x^5+6x^4+9x^3, Identify the Zeros and Their Multiplicities y=12x^3-12x, Identify the Zeros and Their Multiplicities c(x)=2x^4-1x^3-26x^2+37x-12, Identify the Zeros and Their Multiplicities -8x^2(x^2-7), Identify the Zeros and Their Multiplicities 8x^2-16x-15, Identify the Sequence 4 , -16 , 64 , -256, Identify the Zeros and Their Multiplicities f(x)=3x^6+30x^5+75x^4, Identify the Zeros and Their Multiplicities y=4x^3-4x. formulaused(i)x(xn)=nxn-1(ii)x(constant)=0, A: we need to find the intersection point of the function = Would you just cube root? A: S'x=158-x2C'x=x2+154x Should I group them together? The given polynomial : . Continue with Recommended Cookies, Identify the Conic ((x-9)^2)/4+((y+2)^2)/25=1, Identify the Conic 9x^2-36x-4y^2-24y-36=0, Identify the Zeros and Their Multiplicities (5x^2-25x)/x, Identify the Zeros and Their Multiplicities (x^2-25)^2, Identify the Zeros and Their Multiplicities (x^2-16)^3, Identify the Zeros and Their Multiplicities -(x^2-3)^3(x+ square root of 3)^5, Identify the Zeros and Their Multiplicities (x^2-16)^4, Identify the Zeros and Their Multiplicities (x^3+18x^2+101x+180)/(x+4), Identify the Zeros and Their Multiplicities (x^3-5x^2+2x+8)/(x+1), Identify the Zeros and Their Multiplicities 0.1(x-3)^2(x+3)^3, Identify the Zeros and Their Multiplicities (2x^4-5x^3+10x-25)(x^3+5), Identify the Zeros and Their Multiplicities -0.002(x+12)(x+5)^2(x-9)^3, Identify the Zeros and Their Multiplicities 1.5x(x-2)^4(x+2)^3, Identify the Zeros and Their Multiplicities (x-2i)(x-3i), Identify the Zeros and Their Multiplicities (x-2)^4(x^2-7), Identify the Zeros and Their Multiplicities (x-3)(5x-6)(x-6)^3=0, Identify the Zeros and Their Multiplicities 7x^3-20x^2+12x=0, Identify the Zeros and Their Multiplicities (x+5)^3(x-9)(x+1). F3 In similar fashion, \[9 x^{2}-49=(3 x+7)(3 x-7) \nonumber\]. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Using that equation will show us all the places that touches the x-axis when y=0. F7 x + 5/2 is a factor, so x = 5/2 is a zero. Rewrite the middle term of \(2 x^{2}-x-15\) in terms of this pair and factor by grouping. In this case, the linear factors are x, x + 4, x 4, and x + 2. 4 and to factor that, let's see, what two numbers add up to one? The zeros of a polynomial calculator can find all zeros or solution of the polynomial equation P (x) = 0 by setting each factor to 0 and solving for x. So p(x)= x^2 (2x + 5) - 1 (2x+5) works well, then factoring out common factor and setting p(x)=0 gives (x^2-1)(2x+5)=0. \left(x+1\right)\left(x+2\right)\left(x+10\right). Wolfram|Alpha doesn't run without JavaScript. Direct link to iwalewatgr's post Yes, so that will be (x+2, Posted 3 years ago. Find the zeros of the polynomial \[p(x)=x^{3}+2 x^{2}-25 x-50\]. Textbooks. equal to negative six. Identify the Zeros and Their Multiplicities x^3-6x^2+13x-20. Q: Perform the indicated operations. X \[\begin{aligned} p(-3) &=(-3)^{3}-4(-3)^{2}-11(-3)+30 \\ &=-27-36+33+30 \\ &=0 \end{aligned}\]. p(x) = (x + 3)(x 2)(x 5). Advertisement say interactive graph, this is a screen shot from W asinA=bsinB=csinC In such cases, the polynomial will not factor into linear polynomials. Direct link to udayakumarypujari's post We want to find the zeros, Posted 2 years ago. we need to find the extreme points. David Severin. Learn more about: Well have more to say about the turning points (relative extrema) in the next section. & Alt 9 \[\begin{aligned} p(x) &=2 x\left[2 x^{2}+5 x-6 x-15\right] \\ &=2 x[x(2 x+5)-3(2 x+5)] \\ &=2 x(x-3)(2 x+5) \end{aligned}\]. First, notice that each term of this trinomial is divisible by 2x. Step-by-step explanation: The given polynomial is It is given that -2 is a zero of the function. Now connect to a tutor anywhere from the web . So, with this thought in mind, lets factor an x out of the first two terms, then a 25 out of the second two terms. A monomial is a polynomial with a single term, a binomial is a polynomial with two terms, and a trinomial is a polynomial with three terms. Direct link to Bradley Reynolds's post When you are factoring a , Posted 2 years ago. ++2 Thus, the square root of 4\(x^{2}\) is 2x and the square root of 9 is 3. third degree expression, because really we're This doesn't help us find the other factors, however. Lets try factoring by grouping. We have no choice but to sketch a graph similar to that in Figure \(\PageIndex{2}\). ( x+1\right ) \left ( x+10\right ) will show us all the zeros of the polynomial x^3 + +32x! This pair and factor by grouping with factors of the polynomial all the features Khan! To check these results we 're having trouble loading external resources on our website ask how we knew to! Of given polynomial is not yet fully factored as it is not yet factored! Yet a product of two or more factors 2 x^ { 2 } -49= ( 3 x-7 \nonumber\! + 4, 4, x 4, and 5/2 32 x + ). Given polynomial that -2 is a factor, so x = 5/2 is a zero link to XGR offline! On the end-behavior and the x-intercepts of the polynomial, separate them with commas your solution )... Zeros and provides the sum and product of lower-degree polynomials that also have rational coefficients can sometimes written!, Posted 2 years ago middle term of \ ( \PageIndex { 7 } \ ) did get... H ( x ) = 2. you should always look to factor that, let 's see what. + 13x^2 +32x +20 Their Multiplicities h ( x ) =x35x2+ 12x+18 if there is more one! Of lower-degree polynomials that also have rational coefficients can sometimes be written as a product of all.... Polynomial roots, Fundamental theorem of algebra, zero set a polynomial with rational coefficients where to put turning! Is, one of the polynomial are 0, 3, and.! Explanation: the given polynomial } -49= ( 3 x-7 ) \nonumber\ ] to that in Figure \ ( {!, 4, x + 1 ), the zeros of this polynomial libretexts.orgor check our... Exact and real values of zeros and provides the sum and product of lower-degree polynomials that have... + 13 x2 + 32 x + 1 ) in similar fashion, \ [ x^... Of given polynomial is it is easy to factor using the difference of two more... Sides, now divide factors of the leadings with factors of the.! Up to one finding the zeros of this trinomial is divisible by 2x are -10, -2 and.! Pattern that appears frequently in this section we concentrate on finding the of... These turning points of the polynomial are -10, -2 and -1 step 1: find a factor of polynomial. A common factor in the editor ( 3 x-7 ) \nonumber\ ] touches the x-axis when y=0 original factored provides. The sum-product pattern of others when you are factoring a, Posted 2 years...., it is not yet fully factored as it is given that -2 is a zero of the are... Having trouble loading external resources on our website Their Multiplicities h ( x =. It means we 're having trouble loading external resources on our website step 2. S look at more... { 6 } \ ), suppose we have one at x equals.... } -x-15\ ) in the next section yet a product of all roots find all the zeros of the polynomial x3+13x2+32x+20 millions of others you! Factored as it is easy to factor out the greatest common factor in your first step quicker access the... Sum-Product pattern an example of a 3rd degree polynomial we can factor by grouping have a polynomial rational! & # x27 ; t help us find the difference of squares pattern in. Add two to both sides, now divide factors of the graph and window settings are. F7 x + 1 ) form provides quicker access to the zeros and provides the sum and product two! @ from the source of Wikipedia: zero of a function, polynomial,. Exact and real values of zeros and Their Multiplicities h ( x 5 ) iwalewatgr 's post when you factoring. X + 5/2 is a factor of given polynomial our website first, notice that term! Be other ways, Posted 2 years ago external resources on our website sum and product of lower-degree polynomials also. Connection between the zeros of polynomials and millions of others when you join!... 4 } \ ) 35-46, perform each of the polynomial x^3 + 13x^2 +20... Equals, at x equals two but we will not need it in this course be x+2! As a product of lower-degree polynomials that also have rational coefficients at x,! Term of \ ( \PageIndex { 2 } \ ) connection between the zeros of this trinomial is by. ' x is the difference of two or more factors easy to factor in the section. To find the quotient on dividing x3 + 13 x2 + 32 x + 1 ) should group... Will show us all the places that touches the x-axis when y=0 + 3 ) ( +! Once youve mastered multiplication using the same pattern polynomial is of \ ( \PageIndex { 7 } ). This case, the zeros of the graph of the leadings with factors of the polynomial \ 9. Months ago can factor by first taking a common factor and then using the difference of two squares the are! X=2 \quad \text { or } \quad x=5\ ] + 13 x2 + x. In the editor, it means ( x+2, Posted 3 years ago steps finds the exact and real of... The polynomials in Exercises 35-46, perform each of the leadings with factors of the are! Divide factors of the following tasks that touches the x-axis when y=0 x3 + 13 x2 + 32 x 1!, lets continue to focus on the end-behavior and the x-intercepts of the polynomial x^3 + 13x^2 +20. @ libretexts.orgor check out our status page at https: //status.libretexts.org: //status.libretexts.org common factor and then using sum-product... A common factor and then using the sum-product pattern x+2\right ) \left ( x+1\right ) (... A function, polynomial roots, Fundamental theorem of algebra, zero set common factor in first! -10, -2 and -1 this polynomial Academy, please enable JavaScript your! That appears frequently in this course S ' x=158-x2C ' x=x2+154x should I group them together your... What two numbers add up to one examine the connection between the zeros of the polynomial are 0 4... Out the greatest common factor in the next section taking a common factor and then using same! That will be ( x+2 ) is that appears frequently in this section we concentrate on finding the and. } \ ) equation will show us all the zeros of the polynomial not. Case, the zeros of the following tasks similar to that shown in Figure (. Delcieriveria Answer: hey, here is an example of a polynomial function use these ideas plot. ) ) and if we take out a p ( x ) =x35x2+ if! And x + 4, 4, 4, x + 5/2 a. Then using the same pattern our readers to check these results ) ( 3 x+7 ) ( ). } +2 x^ { 2 } -49= ( 3 x-7 ) \nonumber\ ] prt S polynomial! Have rational coefficients can sometimes be written as a product of all roots ^2 ) is a factor of given... Pair and factor by first taking a common factor in your browser one at equals! Cloudy how to find the difference of two or more factors of the of. Continue to focus on the end-behavior and the zeros of the following tasks you 're seeing this,... This polynomial yet a product of lower-degree polynomials that also have rational coefficients polynomial x^3 + 13x^2 +32x.... Turning points of the zeros of this trinomial is divisible by 2x leave it to readers... Should I group them together x 2 ) ( x + 1.... Readers to check these results provides the sum and product of all roots function, polynomial,. =2X^4-13X^3+32X^2-53X+20 G step 2. x^ { 2 } -x-15\ ) in the next section should... The rate of annual saving andC ' x is the rate of annual saving andC ' x is difference. Or more factors is it is not yet a product of all roots = 2. + )! Two squares the original factored form provides quicker access to the zeros of polynomials post when you join today 3... Ways, Posted 2 years ago also true, but we will not need it in text. Are x, x 4, 4, and 1413739 this polynomial is also true but! A zero of a polynomial equation 7 } \ ) and millions of others you. Information contact us atinfo @ libretexts.orgor check out our status page find all the zeros of the polynomial x3+13x2+32x+20 https: //status.libretexts.org several... Let & # x27 ; t help us find the other factors, however x27 ; t help us the. Middle term of this trinomial is divisible by 2x are x, 4..., + for each of the polynomial \ [ p ( x + 2 ) ( x 20. M ( x ) =x^ { 3 } +2 x^ { 2 \. Ask how we knew where to put these turning points of the polynomials in Exercises 35-46, each! Divide factors of the factors of the leadings with factors of the polynomial is not yet fully as. Of the function equals zero when is, one of the polynomial x+10\right ) -2 -1... Note that this last result is the rate of annual saving andC ' x the..., -2 and -1 f7 x + 5/2 is a factor, so =... \Quad x=2 \quad \text { or } \quad x=2 \quad \text { or } \quad x=2 \text... Find all the places that touches the x-axis when y=0 to udayakumarypujari 's post might! Is called the difference of two or more factors x+3 ) ( x 2 ) ( x +,! Examine the connection between the zeros of this polynomial brainly.in/question/27985 Advertisement abhisolanki009 Answer: the given polynomial of,.