Answer :
Sure! Let's examine each polynomial and its corresponding binomial to determine whether the binomial is a factor of the polynomial. We'll use the idea that if a binomial is a factor of a polynomial, substituting the root of the binomial into the polynomial should result in zero.
### 39. Polynomial: [tex]\( f(x) = 2x^3 + 5x^2 - 37x - 60 \)[/tex], Binomial: [tex]\( x - 4 \)[/tex]
- To check if [tex]\( x - 4 \)[/tex] is a factor, substitute [tex]\( x = 4 \)[/tex] into [tex]\( f(x) \)[/tex].
- Calculating [tex]\( f(4) \)[/tex]:
[tex]\[
f(4) = 2(4)^3 + 5(4)^2 - 37(4) - 60 = 2(64) + 5(16) - 148 - 60 = 128 + 80 - 148 - 60 = 0
\][/tex]
- Since [tex]\( f(4) = 0 \)[/tex], [tex]\( x - 4 \)[/tex] is a factor.
### 40. Polynomial: [tex]\( g(x) = 3x^3 - 28x^2 + 29x + 140 \)[/tex], Binomial: [tex]\( x + 7 \)[/tex]
- Check if [tex]\( x + 7 \)[/tex] is a factor by substituting [tex]\( x = -7 \)[/tex] into [tex]\( g(x) \)[/tex].
- Calculating [tex]\( g(-7) \)[/tex]:
[tex]\[
g(-7) = 3(-7)^3 - 28(-7)^2 + 29(-7) + 140
\][/tex]
[tex]\[
= 3(-343) - 28(49) - 203 + 140 = -1029 - 1372 - 203 + 140 \neq 0
\][/tex]
- Since [tex]\( g(-7) \neq 0 \)[/tex], [tex]\( x + 7 \)[/tex] is not a factor.
### 41. Polynomial: [tex]\( h(x) = 6x^5 - 15x^4 - 9x^3 \)[/tex], Binomial: [tex]\( x + 3 \)[/tex]
- Check if [tex]\( x + 3 \)[/tex] is a factor by substituting [tex]\( x = -3 \)[/tex] into [tex]\( h(x) \)[/tex].
- Calculating [tex]\( h(-3) \)[/tex]:
[tex]\[
h(-3) = 6(-3)^5 - 15(-3)^4 - 9(-3)^3
\][/tex]
[tex]\[
= 6(-243) - 15(81) - 9(-27) = -1458 - 1215 + 243 \neq 0
\][/tex]
- Since [tex]\( h(-3) \neq 0 \)[/tex], [tex]\( x + 3 \)[/tex] is not a factor.
### 42. Polynomial: [tex]\( g(x) = 8x^5 - 58x^4 + 60x^3 + 140 \)[/tex], Binomial: [tex]\( x - 6 \)[/tex]
- Substitute [tex]\( x = 6 \)[/tex] into [tex]\( g(x) \)[/tex] to verify if it's a factor.
- Calculating [tex]\( g(6) \)[/tex]:
[tex]\[
g(6) = 8(6)^5 - 58(6)^4 + 60(6)^3 + 140
\][/tex]
[tex]\[
= 8(7776) - 58(1296) + 60(216) + 140 \neq 0
\][/tex]
- Since [tex]\( g(6) \neq 0 \)[/tex], [tex]\( x - 6 \)[/tex] is not a factor.
### 43. Polynomial: [tex]\( h(x) = 6x^4 - 6x^3 - 84x^2 + 144x \)[/tex], Binomial: [tex]\( x + 4 \)[/tex]
- Substitute [tex]\( x = -4 \)[/tex] into [tex]\( h(x) \)[/tex] to verify if it's a factor.
- Calculating [tex]\( h(-4) \)[/tex]:
[tex]\[
h(-4) = 6(-4)^4 - 6(-4)^3 - 84(-4)^2 + 144(-4)
\][/tex]
[tex]\[
= 6(256) - 6(-64) - 84(16) + 144(-4) = 1536 + 384 - 1344 - 576 = 0
\][/tex]
- Since [tex]\( h(-4) = 0 \)[/tex], [tex]\( x + 4 \)[/tex] is a factor.
### 44. Polynomial: [tex]\( t(x) = 48x^4 + 36x^3 - 138x^2 - 36x \)[/tex], Binomial: [tex]\( x + 2 \)[/tex]
- Substitute [tex]\( x = -2 \)[/tex] into [tex]\( t(x) \)[/tex] to verify if it's a factor.
- Calculating [tex]\( t(-2) \)[/tex]:
[tex]\[
t(-2) = 48(-2)^4 + 36(-2)^3 - 138(-2)^2 - 36(-2)
\][/tex]
[tex]\[
= 48(16) + 36(-8) - 138(4) + 72 = 768 - 288 - 552 + 72 = 0
\][/tex]
- Since [tex]\( t(-2) = 0 \)[/tex], [tex]\( x + 2 \)[/tex] is a factor.
In summary, the binomials [tex]\( x-4 \)[/tex], [tex]\( x+4 \)[/tex], and [tex]\( x+2 \)[/tex] are factors of their respective polynomials. The binomials [tex]\( x+7 \)[/tex], [tex]\( x+3 \)[/tex], and [tex]\( x-6 \)[/tex] are not factors of their respective polynomials.
### 39. Polynomial: [tex]\( f(x) = 2x^3 + 5x^2 - 37x - 60 \)[/tex], Binomial: [tex]\( x - 4 \)[/tex]
- To check if [tex]\( x - 4 \)[/tex] is a factor, substitute [tex]\( x = 4 \)[/tex] into [tex]\( f(x) \)[/tex].
- Calculating [tex]\( f(4) \)[/tex]:
[tex]\[
f(4) = 2(4)^3 + 5(4)^2 - 37(4) - 60 = 2(64) + 5(16) - 148 - 60 = 128 + 80 - 148 - 60 = 0
\][/tex]
- Since [tex]\( f(4) = 0 \)[/tex], [tex]\( x - 4 \)[/tex] is a factor.
### 40. Polynomial: [tex]\( g(x) = 3x^3 - 28x^2 + 29x + 140 \)[/tex], Binomial: [tex]\( x + 7 \)[/tex]
- Check if [tex]\( x + 7 \)[/tex] is a factor by substituting [tex]\( x = -7 \)[/tex] into [tex]\( g(x) \)[/tex].
- Calculating [tex]\( g(-7) \)[/tex]:
[tex]\[
g(-7) = 3(-7)^3 - 28(-7)^2 + 29(-7) + 140
\][/tex]
[tex]\[
= 3(-343) - 28(49) - 203 + 140 = -1029 - 1372 - 203 + 140 \neq 0
\][/tex]
- Since [tex]\( g(-7) \neq 0 \)[/tex], [tex]\( x + 7 \)[/tex] is not a factor.
### 41. Polynomial: [tex]\( h(x) = 6x^5 - 15x^4 - 9x^3 \)[/tex], Binomial: [tex]\( x + 3 \)[/tex]
- Check if [tex]\( x + 3 \)[/tex] is a factor by substituting [tex]\( x = -3 \)[/tex] into [tex]\( h(x) \)[/tex].
- Calculating [tex]\( h(-3) \)[/tex]:
[tex]\[
h(-3) = 6(-3)^5 - 15(-3)^4 - 9(-3)^3
\][/tex]
[tex]\[
= 6(-243) - 15(81) - 9(-27) = -1458 - 1215 + 243 \neq 0
\][/tex]
- Since [tex]\( h(-3) \neq 0 \)[/tex], [tex]\( x + 3 \)[/tex] is not a factor.
### 42. Polynomial: [tex]\( g(x) = 8x^5 - 58x^4 + 60x^3 + 140 \)[/tex], Binomial: [tex]\( x - 6 \)[/tex]
- Substitute [tex]\( x = 6 \)[/tex] into [tex]\( g(x) \)[/tex] to verify if it's a factor.
- Calculating [tex]\( g(6) \)[/tex]:
[tex]\[
g(6) = 8(6)^5 - 58(6)^4 + 60(6)^3 + 140
\][/tex]
[tex]\[
= 8(7776) - 58(1296) + 60(216) + 140 \neq 0
\][/tex]
- Since [tex]\( g(6) \neq 0 \)[/tex], [tex]\( x - 6 \)[/tex] is not a factor.
### 43. Polynomial: [tex]\( h(x) = 6x^4 - 6x^3 - 84x^2 + 144x \)[/tex], Binomial: [tex]\( x + 4 \)[/tex]
- Substitute [tex]\( x = -4 \)[/tex] into [tex]\( h(x) \)[/tex] to verify if it's a factor.
- Calculating [tex]\( h(-4) \)[/tex]:
[tex]\[
h(-4) = 6(-4)^4 - 6(-4)^3 - 84(-4)^2 + 144(-4)
\][/tex]
[tex]\[
= 6(256) - 6(-64) - 84(16) + 144(-4) = 1536 + 384 - 1344 - 576 = 0
\][/tex]
- Since [tex]\( h(-4) = 0 \)[/tex], [tex]\( x + 4 \)[/tex] is a factor.
### 44. Polynomial: [tex]\( t(x) = 48x^4 + 36x^3 - 138x^2 - 36x \)[/tex], Binomial: [tex]\( x + 2 \)[/tex]
- Substitute [tex]\( x = -2 \)[/tex] into [tex]\( t(x) \)[/tex] to verify if it's a factor.
- Calculating [tex]\( t(-2) \)[/tex]:
[tex]\[
t(-2) = 48(-2)^4 + 36(-2)^3 - 138(-2)^2 - 36(-2)
\][/tex]
[tex]\[
= 48(16) + 36(-8) - 138(4) + 72 = 768 - 288 - 552 + 72 = 0
\][/tex]
- Since [tex]\( t(-2) = 0 \)[/tex], [tex]\( x + 2 \)[/tex] is a factor.
In summary, the binomials [tex]\( x-4 \)[/tex], [tex]\( x+4 \)[/tex], and [tex]\( x+2 \)[/tex] are factors of their respective polynomials. The binomials [tex]\( x+7 \)[/tex], [tex]\( x+3 \)[/tex], and [tex]\( x-6 \)[/tex] are not factors of their respective polynomials.
