Answer :
Sure, let's break this down step-by-step for each polynomial given:
### Checking if [tex]\( x+2 \)[/tex] is a factor of [tex]\( x^3-10x^2+6x+60 \)[/tex]
1. Polynomial: [tex]\( x^3 - 10x^2 + 6x + 60 \)[/tex]
2. Potential Factor: [tex]\( x + 2 \)[/tex]
To check if [tex]\( x + 2 \)[/tex] is a factor, we substitute [tex]\( x = -2 \)[/tex] into the polynomial and see if the result is zero:
[tex]\[ (-2)^3 - 10(-2)^2 + 6(-2) + 60 \][/tex]
Calculations:
[tex]\[ -8 - 40 - 12 + 60 = -8 - 52 + 60 = -60 + 60 = 0 \][/tex]
Since the remainder is 0, [tex]\( x + 2 \)[/tex] is indeed a factor of [tex]\( x^3 - 10x^2 + 6x + 60 \)[/tex].
Answer:
[tex]\[ \checkmark \text{ Yes}, \text{ The remainder when you divide is } 0 \][/tex]
### Checking if [tex]\( 2x-7 \)[/tex] is a factor of [tex]\( 8x^3+8x^2+14x-490 \)[/tex]
1. Polynomial: [tex]\( 8x^3 + 8x^2 + 14x - 490 \)[/tex]
2. Potential Factor: [tex]\( 2x - 7 \)[/tex]
To check if [tex]\( 2x - 7 \)[/tex] is a factor, we substitute [tex]\( x = \frac{7}{2} \)[/tex] into the polynomial and see if the result is zero:
[tex]\[ 8 \left(\frac{7}{2}\right)^3 + 8\left(\frac{7}{2}\right)^2 + 14\left(\frac{7}{2}\right) - 490 \][/tex]
Calculations:
[tex]\[ 8 \left(\frac{343}{8}\right) + 8 \left(\frac{49}{4}\right) + 14 \left(\frac{7}{2}\right) - 490 \][/tex]
[tex]\[ = 343 + 98 + 49 - 490 \][/tex]
[tex]\[ = 343 + 147 - 490 = 490 - 490 = 0 \][/tex]
Since the remainder is 0, [tex]\( 2x - 7 \)[/tex] is a factor of [tex]\( 8x^3 + 8x^2 + 14x - 490 \)[/tex].
Answer:
[tex]\[ \checkmark \text{ Yes}, \text{ The remainder when you divide is } 0 \][/tex]
### Checking if [tex]\( x-5 \)[/tex] is a factor of [tex]\( x^5-2x^4-2x^3-5x^2+6x+4280 \)[/tex]
1. Polynomial: [tex]\( x^5 - 2x^4 - 2x^3 - 5x^2 + 6x + 4280 \)[/tex]
2. Potential Factor: [tex]\( x - 5 \)[/tex]
To check if [tex]\( x - 5 \)[/tex] is a factor, we substitute [tex]\( x = 5 \)[/tex] into the polynomial and see if the result is zero:
[tex]\[ 5^5 - 2(5^4) - 2(5^3) - 5(5^2) + 6(5) + 4280 \][/tex]
Calculations:
[tex]\[ 3125 - 2(625) - 2(125) - 5(25) + 30 + 4280 \][/tex]
[tex]\[ = 3125 - 1250 - 250 - 125 + 30 + 4280 \][/tex]
[tex]\[ = 3125 - 1625 + 4310 \][/tex]
[tex]\[ = 1500 + 4310 = 5810 \][/tex]
Since the remainder is 5810, [tex]\( x - 5 \)[/tex] is not a factor of [tex]\( x^5 - 2x^4 - 2x^3 - 5x^2 + 6x + 4280 \)[/tex].
Answer:
[tex]\[ \text{No}, \text{ The remainder when you divide is } 5810 \][/tex]
I hope this step-by-step solution helps you understand the process of determining whether a given polynomial factor is indeed a factor of the polynomial!
### Checking if [tex]\( x+2 \)[/tex] is a factor of [tex]\( x^3-10x^2+6x+60 \)[/tex]
1. Polynomial: [tex]\( x^3 - 10x^2 + 6x + 60 \)[/tex]
2. Potential Factor: [tex]\( x + 2 \)[/tex]
To check if [tex]\( x + 2 \)[/tex] is a factor, we substitute [tex]\( x = -2 \)[/tex] into the polynomial and see if the result is zero:
[tex]\[ (-2)^3 - 10(-2)^2 + 6(-2) + 60 \][/tex]
Calculations:
[tex]\[ -8 - 40 - 12 + 60 = -8 - 52 + 60 = -60 + 60 = 0 \][/tex]
Since the remainder is 0, [tex]\( x + 2 \)[/tex] is indeed a factor of [tex]\( x^3 - 10x^2 + 6x + 60 \)[/tex].
Answer:
[tex]\[ \checkmark \text{ Yes}, \text{ The remainder when you divide is } 0 \][/tex]
### Checking if [tex]\( 2x-7 \)[/tex] is a factor of [tex]\( 8x^3+8x^2+14x-490 \)[/tex]
1. Polynomial: [tex]\( 8x^3 + 8x^2 + 14x - 490 \)[/tex]
2. Potential Factor: [tex]\( 2x - 7 \)[/tex]
To check if [tex]\( 2x - 7 \)[/tex] is a factor, we substitute [tex]\( x = \frac{7}{2} \)[/tex] into the polynomial and see if the result is zero:
[tex]\[ 8 \left(\frac{7}{2}\right)^3 + 8\left(\frac{7}{2}\right)^2 + 14\left(\frac{7}{2}\right) - 490 \][/tex]
Calculations:
[tex]\[ 8 \left(\frac{343}{8}\right) + 8 \left(\frac{49}{4}\right) + 14 \left(\frac{7}{2}\right) - 490 \][/tex]
[tex]\[ = 343 + 98 + 49 - 490 \][/tex]
[tex]\[ = 343 + 147 - 490 = 490 - 490 = 0 \][/tex]
Since the remainder is 0, [tex]\( 2x - 7 \)[/tex] is a factor of [tex]\( 8x^3 + 8x^2 + 14x - 490 \)[/tex].
Answer:
[tex]\[ \checkmark \text{ Yes}, \text{ The remainder when you divide is } 0 \][/tex]
### Checking if [tex]\( x-5 \)[/tex] is a factor of [tex]\( x^5-2x^4-2x^3-5x^2+6x+4280 \)[/tex]
1. Polynomial: [tex]\( x^5 - 2x^4 - 2x^3 - 5x^2 + 6x + 4280 \)[/tex]
2. Potential Factor: [tex]\( x - 5 \)[/tex]
To check if [tex]\( x - 5 \)[/tex] is a factor, we substitute [tex]\( x = 5 \)[/tex] into the polynomial and see if the result is zero:
[tex]\[ 5^5 - 2(5^4) - 2(5^3) - 5(5^2) + 6(5) + 4280 \][/tex]
Calculations:
[tex]\[ 3125 - 2(625) - 2(125) - 5(25) + 30 + 4280 \][/tex]
[tex]\[ = 3125 - 1250 - 250 - 125 + 30 + 4280 \][/tex]
[tex]\[ = 3125 - 1625 + 4310 \][/tex]
[tex]\[ = 1500 + 4310 = 5810 \][/tex]
Since the remainder is 5810, [tex]\( x - 5 \)[/tex] is not a factor of [tex]\( x^5 - 2x^4 - 2x^3 - 5x^2 + 6x + 4280 \)[/tex].
Answer:
[tex]\[ \text{No}, \text{ The remainder when you divide is } 5810 \][/tex]
I hope this step-by-step solution helps you understand the process of determining whether a given polynomial factor is indeed a factor of the polynomial!
