a. Answer The depressed polynomial is − 9 x 4 + 50 x 3 + 51 x 2 − 150 x − 72 . Explanation Given information: f ( x ) = − 9 x 5 + 104 x 4 − 249 x 3 − 456 x 2 + 828 x + 432 Formula used: Synthetic Substitution :- Synthetic division is used to find the value of a function. Factor Theorem :- The binomial x − r is a factor of the polynomial P ( x ) if and only if P ( r ) = 0 . Remainder Theorem:- If a polynomial P ( x ) is divided by x − r , the remainder s a constant P ( r ) , and P ( x ) = Q ( x ) ( x − r ) + P ( r ) , Where Q ( x ) is a polynomial with degree one less than P ( x ) . Calculation: Consider , f ( x ) = − 9 x 5 + 104 x 4 − 249 x 3 − 456 x 2 + 828 x + 432 Since, the binomial x − 6 is a factor of the polynomial by using Synthetic division we get, 6 − 9 104 − 249 − 456 828 43 2 − 5 4 300 306 − 900 − 432 − 9 50 51 − 150 − 72 0 Thus, the remainder is zero. The polynomial − 9 x 5 + 104 x 4 − 249 x 3 − 456 x 2 + 828 x + 432 can be factored as ( x − 6 ) ( − 9 x 4 + 50 x 3 + 51 x 2 − 150 x − 72 ) . The polynomial − 9 x 4 + 50 x 3 + 51 x 2 − 150 x − 72 is the depressed polynomial. b. To determine To calculate: the values for the depressed polynomial. Answer Values of x Value − 5 − 9 , 922 − 4 − 4 , 160 − 3 − 1 , 242 − 2 − 112 − 1 70 0 − 72 1 − 130 2 88 3 558 4 1 , 040 5 1 , 078 6 0 Explanation Given information: f ( x ) = − 9 x 5 + 104 x 4 − 249 x 3 − 456 x 2 + 828 x + 432 − 9 x 4 + 50 x 3 + 51 x 2 − 150 x − 72 is the depressed polynomial. − 5 ≤ x ≤ 6 value of x . Formula used: Direct Substitution :- The method is used to find the value of a function. Calculations: Values of x Depressed polynomial − 9 x 4 + 50 x 3 + 51 x 2 − 150 x − 72 Value − 5 = − 9 ( − 5 ) 4 + 50 ( − 5 ) 3 + 51 ( − 5 ) 2 − 150 ( − 5 ) − 72 = − 5625 − 6250 + 1275 + 750 − 72 = − 9922 − 9 , 922 − 4 = − 9 ( − 4 ) 4 + 50 ( − 4 ) 3 + 51 ( − 4 ) 2 − 150 ( − 4 ) − 72 = − 2304 − 3200 + 816 + 600 − 72 = − 4160 − 4 , 160 − 3 = − 9 ( − 3 ) 4 + 50 ( − 3 ) 3 + 51 ( − 3 ) 2 − 150 ( − 3 ) − 72 = − 729 − 1350 + 459 + 450 − 72 = − 1242 − 1 , 242 − 2 = − 9 ( − 2 ) 4 + 50 ( − 2 ) 3 + 51 ( − 2 ) 2 − 150 ( − 2 ) − 72 = − 144 − 400 + 204 + 300 − 72 = − 112 − 112 − 1 = − 9 ( − 1 ) 4 + 50 ( − 1 ) 3 + 51 ( − 1 ) 2 − 150 ( − 1 ) − 72 = − 9 − 50 + 51 + 150 − 72 = 70 70 0 = − 9 ( 0 ) 4 + 50 ( 0 ) 3 + 51 ( 0 ) 2 − 150 ( 0 ) − 72 = − 0 − 0 + 0 + 0 − 72 = − 72 − 72 1 = − 9 ( 1 ) 4 + 50 ( 1 ) 3 + 51 ( 1 ) 2 − 150 ( 1 ) − 72 = − 9 + 50 + 51 − 150 − 72 = − 130 − 130 2 = − 9 ( 2 ) 4 + 50 ( 2 ) 3 + 51 ( 2 ) 2 − 150 ( 2 ) − 72 = − 144 + 400 + 204 − 300 − 72 = 88 88 3 = − 9 ( 3 ) 4 + 50 ( 3 ) 3 + 51 ( 3 ) 2 − 150 ( 3 ) − 72 = − 729 + 1350 + 459 − 450 − 72 = 558 558 4 = − 9 ( 4 ) 4 + 50 ( 4 ) 3 + 51 ( 4 ) 2 − 150 ( 4 ) − 72 = − 2304 + 3200 + 816 − 600 − 72 = 1040 1 , 040 5 = − 9 ( 5 ) 4 + 50 ( 5 ) 3 + 51 ( 5 ) 2 − 150 ( 5 ) − 72 = − 5625 + 6250 + 1275 − 750 − 72 = 1078 1 , 078 6 = − 9 ( 6 ) 4 + 50 ( 6 ) 3 + 51 ( 6 ) 2 − 150 ( 6 ) − 72 = − 11664 + 10800 + 1836 − 900 − 72 = 0 0 The table shows the value of the depressed polynomial . c. To determine To comment: about the locations of the other zeros based on table. Explanation Given information: Values of x Value − 5 − 9 , 922 − 4 − 4 , 160 − 3 − 1 , 242 − 2 − 112 − 1 70 0 − 72 1 − 130 2 88 3 558 4 1 , 040 5 1 , 078 6 0 Formula used: Direct Substitution :- The method is used to find the value of a function. Factor Theorem :- The binomial x − r is a factor of the polynomial P ( x ) if and only if P ( r ) = 0 . Proof: Observing the table we concluded that - With the change in sign there exists a zero between values of x = − 2 and x = − 1 ; x = − 1 and x = 0 . Also at value of x = 6 the value is zero. d. To determine To graph: the depressed polynomial. Explanation Given information: f ( x ) = − 9 x 5 + 104 x 4 − 249 x 3 − 456 x 2 + 828 x + 432 Graph: Figure 1. Interpretation: The above graph shows the conclusion made in part (b) and (c) both. It defines the zero and the curves defines the change in sign of the value of the function.

1st Edition

ISBN: 9780076639908

Author: McGraw-Hill Glencoe

Publisher: MCG

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Question

Chapter 5.6, Problem 31PPS

To determine

To calculate: the depressed polynomial.

Expert Solution

Answer to Problem 31PPS

The depressed polynomial is −9x4+50x3+51x2−150x−72 .

Explanation of Solution

Given information:

f(x)=−9x5+104x4−249x3−456x2+828x+432

Formula used:

Synthetic Substitution :- Synthetic division is used to find the value of a function.

Factor Theorem :- The binomial x−r is a factor of the polynomial P(x) if and only if P(r)=0 .

Remainder Theorem:- If a polynomial P(x) is divided by x−r , the remainder s a constant P(r) , and

P(x)=Q(x)(x−r)+P(r) ,

Where Q(x) is a polynomial with degree one less than P(x) .

Calculation:

Consider ,

f(x)=−9x5+104x4−249x3−456x2+828x+432

Since, the binomial x−6 is a factor of the polynomial by using Synthetic division we get,

6−9 104 −249 −456 828 432 −54 300 306−900−432 −9 50 51 −150−72 0

Thus, the remainder is zero. The polynomial −9x5+104x4−249x3−456x2+828x+432 can be factored as (x−6)(−9x4+50x3+51x2−150x−72) .

The polynomial −9x4+50x3+51x2−150x−72 is the depressed polynomial.

To determine

To calculate: the values for the depressed polynomial.

Expert Solution

Answer to Problem 31PPS

Values of x	Value
−5	−9,922
−4	−4,160
−3	−1,242
−2	−112
−1	70
0	−72
1	−130
2	88
3	558
4	1,040
5	1,078
6	0

Explanation of Solution

Given information:

f(x)=−9x5+104x4−249x3−456x2+828x+432

−9x4+50x3+51x2−150x−72 is the depressed polynomial.

−5≤x≤6 value of x .

Formula used:

Direct Substitution :- The method is used to find the value of a function.

Calculations:

Values of x	Depressed polynomial −9x4+50x3+51x2−150x−72	Value
−5	=−9 (−5) 4 +50 (−5) 3 +51 (−5) 2 −150(−5)−72=−5625−6250+1275+750−72=−9922	−9,922
−4	=−9 (−4) 4 +50 (−4) 3 +51 (−4) 2 −150(−4)−72=−2304−3200+816+600−72=−4160	−4,160
−3	=−9 (−3) 4 +50 (−3) 3 +51 (−3) 2 −150(−3)−72=−729−1350+459+450−72=−1242	−1,242
−2	=−9 (−2) 4 +50 (−2) 3 +51 (−2) 2 −150(−2)−72=−144−400+204+300−72=−112	−112
−1	=−9 (−1) 4 +50 (−1) 3 +51 (−1) 2 −150(−1)−72=−9−50+51+150−72=70	70
0	=−9 (0) 4 +50 (0) 3 +51 (0) 2 −150(0)−72=−0−0+0+0−72=−72	−72
1	=−9 (1) 4 +50 (1) 3 +51 (1) 2 −150(1)−72=−9+50+51−150−72=−130	−130
2	=−9 (2) 4 +50 (2) 3 +51 (2) 2 −150(2)−72=−144+400+204−300−72=88	88
3	=−9 (3) 4 +50 (3) 3 +51 (3) 2 −150(3)−72=−729+1350+459−450−72=558	558
4	=−9 (4) 4 +50 (4) 3 +51 (4) 2 −150(4)−72=−2304+3200+816−600−72=1040	1,040
5	=−9 (5) 4 +50 (5) 3 +51 (5) 2 −150(5)−72=−5625+6250+1275−750−72=1078	1,078
6	=−9 (6) 4 +50 (6) 3 +51 (6) 2 −150(6)−72=−11664+10800+1836−900−72=0	0

The table shows the value of the depressed polynomial .

To determine

To comment: about the locations of the other zeros based on table.

Expert Solution

Explanation of Solution

Given information:

Values of x	Value
−5	−9,922
−4	−4,160
−3	−1,242
−2	−112
−1	70
0	−72
1	−130
2	88
3	558
4	1,040
5	1,078
6	0

Formula used:

Direct Substitution :- The method is used to find the value of a function.

Factor Theorem :- The binomial x−r is a factor of the polynomial P(x) if and only if P(r)=0 .

Proof:

Observing the table we concluded that -

With the change in sign there exists a zero between values of x=−2 and x=−1 ; x=−1 and x=0 . Also at value of x=6 the value is zero.

To determine

To graph: the depressed polynomial.

Expert Solution

Explanation of Solution

Given information:

f(x)=−9x5+104x4−249x3−456x2+828x+432

Graph:

Glencoe Algebra 2 Student Edition C2014, Chapter 5.6, Problem 31PPS

Figure 1.

Interpretation:

The above graph shows the conclusion made in part (b) and (c) both. It defines the zero and the curves defines the change in sign of the value of the function.

Chapter 5.6, Problem 30PPS

Chapter 5.6, Problem 32PPS

Chapter 5 Solutions

Glencoe Algebra 2 Student Edition C2014

Ch. 5.1 - Prob. 1AGP Ch. 5.1 - Prob. 1BGP Ch. 5.1 - Prob. 1CGP Ch. 5.1 - Prob. 1DGP Ch. 5.1 - Prob. 2AGP Ch. 5.1 - Prob. 2BGP Ch. 5.1 - Prob. 3AGP Ch. 5.1 - Prob. 3BGP Ch. 5.1 - Prob. 4AGP Ch. 5.1 - Prob. 4BGP

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