Holt Mcdougal Larson Algebra 2: Student Edition 2012
1st Edition
ISBN: 9780547647159
Author: HOLT MCDOUGAL
Publisher: HOLT MCDOUGAL
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Expert Solution & Answer
Chapter 2.7, Problem 32E
Solution
To describe and correct: The error in writing a polynomial function f with rational coefficients and zeros 2 and 1+i .
The required polynomial function would be f(x)=x3−4x2+6x−4 .
Given information:
Incorrect process of writing a polynomial function:
f(x)=(x−2)[x−(1+i)]=x(x−1−i)−2(x−1−i)=x2−x−ix−2x+2+2i=x2−(3+i)x+(2+2i)
Formula used:
- Factor Theorem: If x=a, b, c are zeros of the function f(x) , then (x−a), (x−b) and (x−c) would be factors of the function.
- Complex Conjugate Theorem: For any polynomial function f(x) with real coefficients, if (a+bi) is an imaginary zero of the function, then (a−bi) will be the other zero of f(x) because complex zeros come in pairs.
- As per Irrational conjugates Theorem, if (a+√b) is a zero of polynomial function f(x) , then (a−√b) will be other zero of the function.
Calculation:
For the given problem, there are 2 zeros. As per complex conjugate theorem, the other zero would be 1−i . This means there would be three zeros for the given problem, while in the given process only two zeros has been used.
Using the factor theorem, the factored form of f would be:
f(x)=(x−2)[x−(1+i)][x−(1−i)]=(x−2)(x−1−i)(x−1+i)
Now expand the factors using difference of squares formula (a−b)(a+b)=a2−b2 :
f(x)=(x−2)(x−1−i)(x−1+i)=(x−2)((x−1)2−i2)=(x−2)((x−1)2+1)(Using identity −i2=1)=(x−2)(x2−2x+1+1)(Using formula (a−b)2=a2−2ab+b2)=(x−2)(x2−2x+2)=x(x2−2x+2)−2(x2−2x+2)(Using distributive property)=x3−2x2+2x−2x2+4x−4=x3−4x2+6x−4
Therefore, the required polynomial function would be f(x)=x3−4x2+6x−4 .
Chapter 2 Solutions
Holt Mcdougal Larson Algebra 2: Student Edition 2012
Ch. 2.1 - Prob. 1GPCh. 2.1 - Prob. 2GPCh. 2.1 - Prob. 3GPCh. 2.1 - Prob. 4GPCh. 2.1 - Prob. 5GPCh. 2.1 - Prob. 6GPCh. 2.1 - Prob. 7GPCh. 2.1 - Prob. 8GPCh. 2.1 - Prob. 1ECh. 2.1 - Prob. 2E
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- 1. For the following subsets of R3, explain whether or not they are a subspace of R³. (a) (b) 1.1 0.65 U = span -3.4 0.23 0.4 -0.44 0 (})} a V {(2) | ER (c) Z= the points in the z-axisarrow_forwardSolve the following equation forx. leave answer in Simplified radical form. 5x²-4x-3=6arrow_forwardMATCHING LIST Question 6 Listen Use the given equations and their discriminants to match them to the type and number of solutions. 00 ed two irrational solutions a. x²+10x-2=-24 two rational solutions b. 8x²+11x-3=7 one rational solution c. 3x²+2x+7=2 two non-real solutions d. x²+12x+45 = 9 DELL FLOWER CHILD 10/20 All Changes S $681 22991arrow_forward
- 88 MULTIPLE CHOICE Question 7 Listen The following irrational expression is given in unsimplified form with four op- tions in simplified form. Select the correct simplified form. Select only one option. A 2±3√√2 B 4±√3 2±√ √3 D 1±√√3 DELL FLOWER CHILD 11/200 4 ± √48 4 ✓ All Changes Saved 165arrow_forwardUse the graph of y = f(x) to answer the following. 3- 2 -4 -2 -1 1 2 3 4 -1 2 m -3- + (d) Find all x for which f(x) = -2. If there is more than one value, separate them with commas or write your answer in interval notation, if necessary. Select "None", if applicable. Value(s) of x for which f(x)=-2: | (0,0) (0,0) (0,0) (0,0) 0,0... -00 None (h) Determine the range of f. The range is (0,0) Garrow_forwardWhat is g(f(4))arrow_forward
- 10) Multiply (8m + 3)² A) 8m²+11m+6 B) m² + 48m+9 C) 64m²+48m+9 D) 16m²+11m+6arrow_forwardLet R be field and X= R³/s Vector space over R M=(a,b,c)labic, e Rra+b= 3- <3 Show that Ms and why with proof. 1) is convexset and affine set of botost ii) is blanced set and symmetirs set of x iii) is hy per space and hyper plane ofx or hot iii) find f:MR st kerf = M 18/103 and finnd fiM→R/{0} st M= {xEX, f(t) = x, texiαER? jiii) show that Mis Maxsubspace or not and Mis a max. affine set or not.arrow_forwardFind The partial fraction decomposition for each The following 2× B) (x+3) a 3 6 X-3x+2x-6arrow_forward
- 1) Find the partial feraction decomposition for each of 5- X 2 2x+x-1 The following: 3 B) 3 X + 3xarrow_forwardT={(−7,1),(1,−1),(6,−8),(2,8)} Find the domain and range of the inverse. Express your answer as a set of numbers.arrow_forwardT={(−7,1),(1,−1),(6,−8),(2,8)}. Find the inverse. Express your answer as a set of ordered pairs.arrow_forward
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