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Since this quadratic will not factor easily, we will use the Quadratic Formula to find them: Simplifying: This makes the 3 roots: 3, 2+i and 2-i. Consider the form . You can specify conditions of storing and accessing cookies in your browser, given that,root 2 is the zero of the cubic polynomial 6x^3+root 2 x^2-10x-4root^2, find it's other zeroes, Elimination using Addition and Subtractionx - y = 1 ; x + y = 3​, Find xwhere angle ABO=30° and angle ACO=50° ​, please inbox me[tex] {y {12 \frac{ \frac{ \sqrt{ \sqrt{ \sqrt{ \sqrt{ \sqrt{?} If one zero of the polynomial (a^2+9)x^2 + 13x + 6a is reciprocal of the other, find the value of a. The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient In this case, the Leading Coefficient is 6 and the Trailing Constant is -3. Given that root 2 is a zero of the cubic polynomial 6x3 +  √2 x2 – 10x – 4  √2 , find  its other two zeroes. The equation becomes this: (x^3-2x^2)-(2x^2-x-6). So this is a zero of our polynomial, and let's see, so x to the third, so we could say, if we subtract five from both sides, we have x to the third is equal to the is equal to negative five, and so if we take both to the one third power we could say x is equal to cube root of negative five. Solve the equation where the result equals to 0. (-6,6) C. (these are the only four numbers with p dividing 2 and q dividing 1). A real number t is called a zero of a polynomial if the value of f(t) = 0 For example f(x) = x^2 â 6x +8 zeros of this equation are 2 and 4 because f(2)= 2^2 -6*2 + 8 = 0 f(4)= 4^2 â 6*4 + 8 =0 . By division algorithm, we have It is given that f(a) = x 4 â 6x 3 + 16x 2 â 25x + 10, when divided by x 2 â 2x + k leaves x + a as remainder. Click hereðto get an answer to your question ï¸ If x = - 9 is a root of x & 3 & 7 2 & x & 2 7 & 6 & x = 0 then find its other two roots a and b , and then find ab . A polynomial can be typified into four based on the number of terms it possesses: monomial, binomial, trinomial or a multinomial. Polynomials: A polynomial is a form of algebraic expression whose every term can either be a constant, a variable, or a product of a constant and variable in which the variable has a whole number or a non-negative number exponent. This is a polynomial of degree 3. Answers (2) Home » â­ï¸ Mathematics » Yuri thinks that 3/4 is a root of the following function. Since the remainder is not zero, then the Factor Theorem says that: x â 1 is not a factor of f (x). If one complex number is root of a polynomial, then the conjugate of the number is also be the root. First observe that the sum of the coefficients is 0, so x=1 is a root. So this is true. Please show work and include any necessary explanations. RD Sharma solutions for Class 10 Maths chapter 2 (Polynomials) include all questions with solution and detail explanation. Consider a quadratic function with two zeros, $x=\frac{2}{5}$ and $x=\frac{3}{4}$. (x + 4)(x - 6) B. …, each day find the total number of pages typewriter will type in 6 months​. If i is a zero of the polynomial, then the factor is (x - i). Use synthetic division to test the possible rational roots and findâ¦ This is a polynomial of degree 3. What have we shown? If the polynomial x 4 â 6x 3 + 16x 2 â 25x + 10 is divided by another polynomial x 2 - 2x + k, the remainder comes out to be x + a, find âkâ and âaâ. …, construct a rectangle ABCD in which side AB=6 and diagonal AC=7.5​. See the answer. The company currently produces 2 million accessories and makes a â¦ [ Start studying 4.11 Algebra 2 Test Answer Check Sheet. For example, we split it into -2x^2-2x^2. If the equation x 3 + q x 2 + r x + s = 0 has roots a 1 , b 1 and c 1 , then the value of (q + r + s) is equal to } } } } }{?} (x + 6)(x - 4) C. (x + 2)(x - 12) D. (x + 12)(x - 2) 2, x^2 - 36 = 0 A. Use the rational root theorem to list all possible rational roots for â¦ 45 Example. That is, I followed the practice used with long division, and wrote the polynomial as x 3 + 0 x 2 â¦ Answers: 1 on a question: If two zeroes of the polynomial p(x) =x4-6x3 -26x2+138x-35 are -â3+2, â3+2. Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of So this means that Note: the term was formed by the first four values in the bottom row. Expert Answer . Ex 2.4, 4 If two zeroes of the polynomial x4 â 6x3 â 26x2 + 138x â 35 are 2 ± â3, find other zeroes. Now that you have , you simply find the possible rational zeros for and test to see which ones are really zeros (ie repeat the first two steps). Ex 2.4, 4 If two zeroes of the polynomial x4 â 6x3 â 26x2 + 138x â 35 are 2 ± â3, find other zeroes. Find real zeros of f(x) 6x3 - 4x2 3x 2. If the polynomial is divided by x â k, the remainder may be found quickly by evaluating the polynomial function at k, that is, f(k). If you have studied complex numbers, then you may see a problem of the following type. Here again is the problem: 3.3 Find roots (zeroes) of : F(x) = 6x 3-7x 2-9x-2 Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0 Rational Roots Test is one of the above mentioned tools. It also show that the other factor is . We can now use polynomial division to evaluate polynomials using the Remainder Theorem. 4.4 Find roots (zeroes) of : F(x) = x 3 + 6x 2 + 6x - 4 Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0 Rational Roots Test is one of the above mentioned tools. The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction $\frac{p}{q}$, where p is a factor of the trailing constant and â¦ This will clear students doubts about any question and improve application skills while preparing for board exams. So, in this section weâll look at a process using the Rational Root Theorem that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes. List all possible rational roots. Thank you! Use the rational root theorem to list all possible rational roots for the equation. Let a, b and c be three distinct real roots of the cubic x 3 + 2 x 2 â 4 x â 4 = 0. By â¦ Factoring is a useful way to find rational roots (which correspond to linear factors) and simple roots involving square roots of integers (which correspond to quadratic factors). In this problem we have . If you have studied complex numbers, then you may see a problem of the following type. Free Rational Roots Calculator - find roots of polynomials using the rational roots theorem step-by-step This website uses cookies to ensure you get the best experience. Sum and product of root of quadratic equation :- For a equation ax^2 + bx + c = 0 , if root â¦ Recall that the Division Algorithm states that given a polynomial dividend f(x) and a non-zero polynomial divisor d(x) where the degree of d(x) is less than or equal to the degree of f(x), there exist uâ¦ Set equal to . The Rational Zero Theorem helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial. Show transcribed image text. If √2 is a zero of the cubic polynomial 6x3 + √2x2 – 10x – 4√2, the find its other two zeroes. Therefore view the full answer. X^3+2x-9=0. Introduction 2 is a factor of 6 3 is a factor of 6 So, 2 × 3 is also a factor of 6 We will use the same in our question Ex2.3, 3 Obtain all other zeroes of 3x4 + 6x3 In the following ,determine whether the given values of x are zeroes of the given polynomial or not x2 + √2x - 4 ; x = √2 , x = √2 , x = -2√2 . If the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a,a + b and a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial. The factor(s) are: of the Leading Coefficient : 1,2 â¦ We can now check each one of them: Consequently, the polynomial has 2 rational roots: x=1 and x=-1 are the only rational zeros of the polynomial P(x). Find a pair of integers whose product is and whose sum is . Solve, a. Use the rational root theorem to list all possible rational roots for the equation. (x-2)(x-3)(x+1) It is usually really, really hard to factorize a cubic function. We are looking for a solution of the given equation, that is, a number c between 1 and 2 such that f(c) = , b = 2 , and N = 0 in the Intermediate Value Theorem. Ex2.3, 3 Obtain all other zeroes of 3x4 + 6x3 â 2x2 â 10x - 5 , if two of its zeroes are â(5/3) and -â(5/3) . It turns out that the possible roots for are: 1, 2, -1, -2 To find zeros for polynomials of degree 3 or higher we use Rational Root Test. EXAMPLE 10 Show that there is a root of the equation 6x3 â 12x2 + 3x â 2 = 0 between 1 and 2. . Question 1047788: Obtain all other zeroes of 3x4 + 6x3 â 2x2 â 10x â 5, if two of its zeroes are â(5/3) and - â(5/3). F(x)-x4-6x3 +8x2 +30x 65; 3 -2i Is A Zero Select One: This problem has been solved! Therefore view the full answer. If a+bi is a zero (root) then a-bi is also a zero of the function. Factor using the rational roots test. Then divide by (x-1) to get a quadratic that is easier to factor and thus solve to find two other roots x=3 and x=-2. Ex: $\sqrt [3]{8} = (8/2)/2$ Is the zeroth root even defined and if so what is $\sqrt [0]{x}$ If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient. }{?} If the remainder is zero, then x = 1 is a zero of x 3 â 1. If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. }^{?} Ex2.3, 3 Obtain all other zeroes of 3x4 + 6x3 â 2x2 â 10x - 5 , if two of its zeroes are â(5/3) and -â(5/3) . There are only roots close to x=-1/2 and close to x=2/3.This reduces our list of candidates to just two; plugging these values into the polynomial, we see that P(-1/2)=0 and P(2/3)=0, so both are indeed rational zeros. We try values for splitting the term -4x^2. (1 + square root of 3, 1 - square root of 3) B. For Î² = 1/2, the above satisfies; so one root Î² = 1/2. Solve for . Also, be careful when you write fractions: 1/x^2 ln(x) is 1/x^2 ln(x), and 1/(x^2 ln(x)) is 1/(x^2 ln(x)). Well, we can also divide polynomials.f(x) ÷ d(x) = q(x) with a remainder of r(x)But it is better to write it as a sum like this: Like in this example using Polynomial Long Division:But you need to know one more thing:Say we divide by a polynomial of degree 1 (such as \"xâ3\") the remainder will have degree 0 (in other words a constant, like \"4\").We will use that idea in the \"Remainder Theorem\": Polynomials with rational coefficients always have as many roots, in the complex plane, as their degree; however, these roots are often not rational numbers. All roots will be less than 1, which eliminates 2 as a possibility. The other 2 â¦ lf x= 4/3 is a root of the polynomial f(x)= 6x^3 - 11x^2+kx- 20, find the value of k. ... Ministry of Education organizes NEP Transforming India Quiz 2020 from 5th Sept to 25th Oct. Know NEP 2020 India Quiz participation rules & winning criteria. Because of the conjugate rule, it is understood that one other root of the polynomial HAS to be (x + i). This polynomial can then be used to find the ... using the rational roots test. Substituting this Î² = 1/2 in (2) & (3) and eliminating Î³, we get, 6Î±^2 - 7Î± + 4 = 0. solving this Î± = 1 or 4/3; so Î³ = 1/3 or 1/4. So now One root is 3. Further on every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. It turns out that the possible roots for are: 1, 2, -1, -2 Divide x^{3}-4x^{2}+5x-2 by x-1 to get x^{2}-3x+2. Every rational root of P(x) is one of the following 4 choices: . In this case, whose product is and whose sum is . If two zeroes of the polynomial p(x) = x4 - 6x3 -26x2 +138x -35 are 2 +√3, find the other zeroes. The other two roots will come from . Obtain all other zeroes of 3x4 + 6x3 â 2x2 â 10x â 5, if two of its zeroes are â(5/3) and - â(5/3) Quotient is x² + 2x + 1 = 0 Compare the equation with ax² + bx + c = 0 We get a = 1 ,b = 2, c = 1 To factorize the value we have to find two value which Set equal to . Polynomial Roots Calculator : 2.3 Find roots (zeroes) of : F(x) = x 3-4x 2 +x+6 Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0 Rational Roots Test is one of the above mentioned tools. And so it does indeed seem that f prime of zero is going to be four times zero, it's gonna be zero over three times the cubed root of zero minus one, of negative one. Introduction Factorising 6 2 is a factor of 6 3 is a factor of 6 So, 2 × 3 is also a factor of 6 We will use the same in our question Hence, (x â 2 â â3) × (x â 2 + â3) is also a factor. F(x)-x4-6x3 +8x2 +30x 65; 3 -2i Is A Zero Select One: This problem has been solved! And so this is three times negative one, or zero over negative three, so this is indeed equal to zero. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. given that â2 is a zero of the cubic polynomial 6x+â2xpower2-10x-4â2,find its other two zeroes - 1331555 To find zeros for polynomials of degree 3 or higher we use Rational Root Test. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers However, if we are not able to factor the polynomial we are unable to do that process. In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation + â â + â¯ + = with integer coefficients â and , â .Solutions of the equation are also called roots or zeroes of the polynomial on the left side.. Introduction Factorising 6 2 is a factor of 6 3 is a factor of 6 So, 2 × 3 is also a factor of 6 We will use the same in our question Hence, (x â 2 â â3) × (x â 2 + â3) is also a factor. Factor out the greatest common factor from each group. If one complex number is root of a polynomial, then the conjugate of the number is also be the root. Likewise Nth root is the result of repeated division by a certain divisor before it becomes $1$ or a decimal. \times \frac{?}{?} See the answer. It costs the company $10 to make each accessory. This site is using cookies under cookie policy. Factor the left side of the equation. So that's one other root. 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. Now that you have , you simply find the possible rational zeros for and test to see which ones are really zeros (ie repeat the first two steps). Given that,root 2 is the zero of the cubic polynomial 6x^3+root 2 x^2-10x-4root^2, find it's other zeroes - 3223061 Answer: One other zero of f(x) is (3+6i) Step-by-step explanation: we know that. Write the factored form using these integers. If â2 is a zero of the cubic polynomial 6x3 + â2x2 â 10x â 4â2, the find its other two zeroes. Because of the zero remainder, this shows that (x-3) is a factor abd that 3 is a root. 1.6k views. Is a polynomial with real coefficients Show transcribed image text. Solution for Given: 6x3 + 25x2 - 24x + 5 = 0. Polynomials class 10 important questions. X^3+2x-9=0. Find All The Zeros. The number of times you divide it before it becomes a decimal is the index. q (x) = 6x3 + 19x2 - 15x - 28 Explain to Yuri 3/4 why cannot be a root (You have to perform this last step! Given that root 2 is a zero of the cubic polynomial 6x3 + root2 x2 10x 4 root2 , find its other two zeroes - Math - Polynomials The Conjugate Zeros Theorem states that if a complex number a + bi is a zero of a polynomial with real coefficients then the complex conjugate of that number, which is a - bi, is also a zero of the polynomial. In the last section, we learned how to divide polynomials. Find the Roots (Zeros) f(x)=x^2-6x+8. The price that a company charged for a computer accessory is given by the equation 100-10x^2 where x is the number of accessories that are produced, in millions. In some situations, we may know two points on a graph but not the zeros. b. Question 3. Using the Factor Theorem, verify that x + 4 is a factor of f (x) = 5x 4 + 16x 3 â 15x 2 + 8x + 16. Introduction 2 is a factor of 6 3 is a factor of 6 So, 2 × 3 is also a factor of 6 We will use the same in our question Ex2.3, 3 Obtain all other zeroes of 3x4 + 6x3 }^{?} The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction$ \frac{p}{q} $, where p is a factor of the trailing constant and â¦ asked Feb 9, 2018 in Class X Maths by priya12 (-12,631 points) If â2 is a zero of the cubic polynomial 6x 3 + â2x 2 â 10x â 4â2, the find its other two zeroes. Sum and product of root of quadratic equation :- For a equation ax^2 + bx + c = 0 , if root are Î± and Î² , Roots â¦ Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of So this means that Note: the term was formed by the first four values in the bottom row. Solve for . Or, = (x 2 â 3x â 3)(x â 2) â 13.x 3 â 5x 2 + 3x â 7 is the dividend, x 2 â 3x â3 is the quotient, and â13 is the remainder.. Find the other zeroes. Factor using the AC method. Learn vocabulary, terms, and more with flashcards, games, and other study tools. However, for this polynomial, we can factor by grouping. Therefore we SOLUTION take a = 1 Let f(x) = 6x3 â 12x2 + 3x â 2 = 0. Given that root 2 is a zero of the cubic polynomial 6x3 + â2 x2 â 10x â 4 â2 , find its other two zeroes. Found 2 solutions by Alan3354, Edwin McCravy: Can anyone tell me how to get new version of Brainly in iPhone. To do the initial set-up, note that I needed to leave "gaps" for the powers of x that are not included in the polynomial. find the zeros of a cubic polynomial x3+6x2-x-30 when product of two of its zeroes is -6, If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is -1, then the product of the other two zeroes is, A polynomial x^3+ax^2+bx+c has a zero -1,prove that the product of other two zeroes is b-a+1, Write a quadratic polynomial whose sum of zeroes is √2 and one of the zero is 1/√2. So the roots are, {1, 1/2, 1/3} or {4/3, 1/2, 1/4} As the second set is not in HP, only the first set is considered as solution. The polynomial equation is 1*x^3 - 8x^2 + 25x - 26 = 0. A real number t is called a zero of a polynomial if the value of f(t) = 0 For example f(x) = x^2 â 6x +8 zeros of this equation are 2 and 4 because f(2)= 2^2 -6*2 + 8 = 0 f(4)= 4^2 â 6*4 + 8 =0 . Use the rational root theorem to list all possible rational roots for â¦ Since is a known root, divide the polynomial by to find the quotient polynomial. Previous question Next question Given that root 2 is a zero of the cubic polynomial 6x3 + root2 x2 10x 4 root2 , find its other two zeroes - Math - Polynomials Here is how to do this problem by synthetic division.. First, to use synthetic division, the divisor must be of the first degree and must have the form x â a.In this example, the divisor is x â 2, with a = 2.. write sin x (or even better sin(x)) instead of sinx. We can factorize each of the expressions in the parentheses: x^2(x-2)-(x-2)(2x+3). If one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of then other two zeroes is. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time and product of its zeroes as 4, -9, -17 respectively. Given that root 2 is a zero of the cubic polynomial 6x3+2x2-10x-4root2, find its other two zeroes ... Alegbra 2. Sometimes I see expressions like tan^2xsec^3x: this will be parsed as tan^(2â¦ How do I solve this? By the rational roots test, our possibilities are ; Well test 1 with synthetic division ; It didnt work, but since 1 is positive and the bottom row is all positive, 1 is an upper bound. Given that root 2 is a zero of the cubic polynomial 6x3+2x2-10x-4root2, find its other two zeroes ALGEBRA 1, Factor each expression x^2 - 10x - 24 A. Previous question Next question Explain the process you used to find your solution: 1 - 2i is a zero of f(x) = x4 - 2x3 + 6x2 - 2x + 5." Given that root 2 is a zero of the cubic polynomial 6x3+2x2-10x-4root2, find its other two zeroes ... Alegbra 2. "Using the given zero, find one other zero of f(x). Find All The Zeros. As we saw in the previous section in order to sketch the graph of a polynomial we need to know what itâs zeroes are. On putting the value of root of a polynomial into that polynomial we get 0so, 6(4/3)3 - 11(4/3)2 + k4/3 - 20 = 0â128/9 - 176/9 + k4/3 - 20 = 0â - 48/9 + k4/3 - 20 = 0â - 16/3 + k4/3 - 20 = 0â(4k - 16)/3 = 20â4k - 16 = 60â4k = 76âk = 19 A critical point is at x equals zero. Letâs walk through the proof of the theorem. here if you want to solve for X means you should put right side some value or else you can assume it as zero. Expert Answer . Factor 6x^3+11x^2-3x-2. Given that √2 is a zero of the cubic polynomial 6x3 + √2 x2 – 10x – 4 √2 , find its other two zeroes. If x + 4 is a factor, then (setting this factor equal to zero and solving) x = â4 is a root. Tap for more steps... Group the first two terms and the last two terms. via "remedy" i assume you propose "simplify", considering there is not any " = ' right here. Find real zeros of f(x) 6x3 - 4x2 3x 2. एक ऐसी कौन सी गेम है जो बाहर खेली जाती है और बहुत देर नहीं लेती​, a typewriter types 22 pages each day for three days in a week and for next two he types 34 pages each day and on the last two days he types 40 pages 0 votes . \times \frac{?}{?} Question 3 Polynomial p is given by $$p(x) = x^4 - 2x^3 - 2x^2 + 6x - 3$$ a) Show that x = 1 is a zero of multiplicity 2. b) Find all zeros of p. Find the Roots (Zeros) f(x)=x^3-3x^2+4x-12. so you might be knowing BOD MAS rule means first the bracket must solve ..and then division(D),multiplication(M),addition(A),substraction(S).. so your equation becomes.. 12-24-5x+6 By Factor theorem, x â k is a factor of the polynomial for each root k . 2 solutions by Alan3354, Edwin McCravy: factor 6x^3+11x^2-3x-2 parentheses: (...: if two zeroes for this polynomial can be typified into four based on number! 6X3 â 12x2 + 3x â 2 = 0 is 0, so x=1 a! Polynomial we are unable to do that process solutions for Class 10 Maths chapter 2 ( polynomials ) all. Root k one complex number is root of a polynomial, then the is! This problem has been solved are not able to factor the polynomial by to find the... using rational... X ) =x4-6x3 -26x2+138x-35 are -â3+2, â3+2 the root â¦ RD Sharma solutions Class! May see a problem of the polynomial we are not able to factor polynomial... 1 is a zero Select one: this problem has been solved, find its other two zeroes of function. ( x-2 ) ( x+1 ) it is understood that one other root of a polynomial, may! 1 - square root of the polynomial we are not able to factor the we... Problem: find real zeros of f ( x ) 6x3 - 4x2 2. Coefficients is 0, so this is three times negative one, or over!, it is understood that one other zero of f ( x ) is one of the following type all! + √2x2 – 10x – 4√2, the find its other two zeroes of the type... Times you divide it before it becomes$ 1 $or a multiplication sign, type least. Question: if two zeroes any question and improve application skills while for. Edwin McCravy: factor 6x^3+11x^2-3x-2 on a graph but not the zeros Maths chapter 2 ( polynomials ) include questions... Games, and more with flashcards, games, and more with flashcards, games and. Becomes a decimal is the problem: find real zeros of f ( x 6... This: ( x^3-2x^2 ) - ( x-2 ) ( 2x+3 ), terms and...$ 1 \$ or a decimal be used to find zeros for polynomials degree! Not the zeros and more with flashcards, games, and other study tools root Î² =.... - 6 ) B to make each accessory cubic polynomial 6x3 + -. ) include all questions with solution and detail explanation 4x2 3x 2 RD Sharma solutions for Class Maths! Factorize a cubic function complex number is root of p ( x ) -x4-6x3 +8x2 +30x 65 ; -2i... Expressions in the parentheses: x^2 ( x-2 ) ( x+1 ) it is usually,. So one root Î² = 1/2 all possible rational roots Test four numbers with p 2. Is three times negative one, or zero over negative three, so x=1 is a zero Select one this. Factor out the greatest common factor from each group k is a known root, divide the has. √2X2 – 10x – 4√2, the find its other two zeroes... 2. Which eliminates 2 as a possibility thinks that 3/4 is a factor of the following type 2 ( polynomials include. 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