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, Problem 22T
Solution
To write: A polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros.
The required polynomial function would be f(x)=x4−10x3+32x2−92x+60 .
Given information:
Zeros of a function: 1+3i, 4+√10 .
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: If f is a polynomial function with real coefficients, and (a+bi) is an imaginary zero of f , then (a−bi) is also a zero of f .
- Irrational conjugates Theorem: Suppose f is a polynomial function with real coefficients, and a and b are rational numbers such that √b is irrational. If a+√b is a zero of f , then a+√b is also a zero of f .
Calculation:
For the given problem, there are 2 zeros. As per irrational conjugate theorem and complex conjugate theorem, the other zeros would be 1−3i, 4−√10 .
Using the factor theorem, the factored form of f
would be:
f(x)=(x−(1−3i))(x−(1+3i))(x−(4+√10))(x−(4−√10))=(x−1+3i)(x−1−3i)(x−4−√10)(x−4+√10)
Now expand the factors using the difference of squares (a−b)(a+b)=a2−b2:
f(x)=(x−1+3i)(x−1−3i)(x−4−√10)(x−4+√10)=((x−1)2−(3i)2)((x−4)2−(√10)2)=((x−1)2−9i2)((x−4)2−10)=((x−1)2+9)((x−4)2−10)(Using identity −i2=1)=(x2−2x+1+9)(x2−8x+16−10)(Using rule (a−b)2=a2−2ab+b2)=(x2−2x+10)(x2−8x+6)=x2(x2−8x+6)−2x(x2−8x+6)+10(x2−8x+6) (Using distributive property)=x4−8x3+6x2−2x3+16x2−12x+10x2−80x+60=x4−10x3+32x2−92x+60
Therefore, the required function would be f(x)=x4−10x3+32x2−92x+60 .
Chapter 2 Solutions
Holt Mcdougal Larson Algebra 2: Student Edition 2012
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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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