Thank you for visiting In Exercises 39 44 determine whether the binomial is a factor of the polynomial 39 tex f x 2x 3 5x 2 37x 60 x. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
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.
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