Answer :
To determine which polynomial function has a leading coefficient of 1 and roots [tex]\( (7+1) \)[/tex] and [tex]\( (5-1) \)[/tex] with multiplicity 1, we must evaluate the given information step-by-step.
1. Identify the roots:
- Root 1: [tex]\( 7+1 = 8 \)[/tex]
- Root 2: [tex]\( 5-1 = 4 \)[/tex]
2. Form the polynomial:
- For the polynomial to have these roots and a leading coefficient of 1, it can be expressed in the form:
[tex]\[ f(x) = (x - 8)(x - 4) \][/tex]
3. Expand the polynomial:
[tex]\[ f(x) = (x - 8)(x - 4) = x^2 - 4x - 8x + 32 = x^2 - 12x + 32 \][/tex]
4. Evaluate possible answers:
- [tex]\( f(x) = (x + 7)(x - 1)(x + 5)(x + 1) \)[/tex]:
This polynomial has roots [tex]\( -7, 1, -5, -1 \)[/tex], which do not match [tex]\( 8 \)[/tex] and [tex]\( 4 \)[/tex].
- [tex]\( f(x) = (x - 7)(x - 1)(x - 5)(x + i) \)[/tex]:
This polynomial has roots [tex]\( 7, 1, 5, -i \)[/tex], which do not match [tex]\( 8 \)[/tex] and [tex]\( 4 \)[/tex].
- [tex]\( f(x) = (x - (7-i))(x - (5+i))(x - (7+1))(x - (5-1)) \)[/tex]:
This polynomial has roots [tex]\( 7-i, 5+i, 8, 4 \)[/tex]. Although it includes 8 and 4, it has additional roots [tex]\( 7-i \)[/tex] and [tex]\( 5+i \)[/tex] that are not given as part of the problem statement.
- [tex]\( f(x) = (x + (7-1))(x + (5+1))(x + (7+1))(x + (5-1)) \)[/tex]:
This polynomial has roots [tex]\( -(7-1) = -6 \)[/tex], [tex]\( -(5+1) = -6 \)[/tex], [tex]\( -(7+1) = -8 \)[/tex], and [tex]\( -(5-1) = -4 \)[/tex]. Clearly, none of these roots are 8 or 4.
So, the polynomial with the correct roots [tex]\(8\)[/tex] and [tex]\(4\)[/tex] and a leading coefficient of 1 is:
[tex]\[ f(x) = (x - 8)(x - 4) \][/tex]
However, given the options provided, none of them directly match our simplified form of [tex]\( f(x) = x^2 - 12x + 32 \)[/tex].
Since the correct polynomial structure matches our calculation most closely compared to the provided selections, we recognize there is no exact match among the given options.
If we focus only on matching the roots [tex]\( 8 \)[/tex] and [tex]\( 4 \)[/tex], the best choice under the circumstances would be to treat it as theoretical alignment to:
[tex]\[ f(x) = (x - 8)(x - 4) \][/tex]
and therefore none of the provided polynomials in point do so explicitly.
Thus, in conclusion, from the options provided,
None of the given polynomial options accurately represent the polynomial with roots [tex]\( 8 \)[/tex] and [tex]\( 4 \)[/tex].
1. Identify the roots:
- Root 1: [tex]\( 7+1 = 8 \)[/tex]
- Root 2: [tex]\( 5-1 = 4 \)[/tex]
2. Form the polynomial:
- For the polynomial to have these roots and a leading coefficient of 1, it can be expressed in the form:
[tex]\[ f(x) = (x - 8)(x - 4) \][/tex]
3. Expand the polynomial:
[tex]\[ f(x) = (x - 8)(x - 4) = x^2 - 4x - 8x + 32 = x^2 - 12x + 32 \][/tex]
4. Evaluate possible answers:
- [tex]\( f(x) = (x + 7)(x - 1)(x + 5)(x + 1) \)[/tex]:
This polynomial has roots [tex]\( -7, 1, -5, -1 \)[/tex], which do not match [tex]\( 8 \)[/tex] and [tex]\( 4 \)[/tex].
- [tex]\( f(x) = (x - 7)(x - 1)(x - 5)(x + i) \)[/tex]:
This polynomial has roots [tex]\( 7, 1, 5, -i \)[/tex], which do not match [tex]\( 8 \)[/tex] and [tex]\( 4 \)[/tex].
- [tex]\( f(x) = (x - (7-i))(x - (5+i))(x - (7+1))(x - (5-1)) \)[/tex]:
This polynomial has roots [tex]\( 7-i, 5+i, 8, 4 \)[/tex]. Although it includes 8 and 4, it has additional roots [tex]\( 7-i \)[/tex] and [tex]\( 5+i \)[/tex] that are not given as part of the problem statement.
- [tex]\( f(x) = (x + (7-1))(x + (5+1))(x + (7+1))(x + (5-1)) \)[/tex]:
This polynomial has roots [tex]\( -(7-1) = -6 \)[/tex], [tex]\( -(5+1) = -6 \)[/tex], [tex]\( -(7+1) = -8 \)[/tex], and [tex]\( -(5-1) = -4 \)[/tex]. Clearly, none of these roots are 8 or 4.
So, the polynomial with the correct roots [tex]\(8\)[/tex] and [tex]\(4\)[/tex] and a leading coefficient of 1 is:
[tex]\[ f(x) = (x - 8)(x - 4) \][/tex]
However, given the options provided, none of them directly match our simplified form of [tex]\( f(x) = x^2 - 12x + 32 \)[/tex].
Since the correct polynomial structure matches our calculation most closely compared to the provided selections, we recognize there is no exact match among the given options.
If we focus only on matching the roots [tex]\( 8 \)[/tex] and [tex]\( 4 \)[/tex], the best choice under the circumstances would be to treat it as theoretical alignment to:
[tex]\[ f(x) = (x - 8)(x - 4) \][/tex]
and therefore none of the provided polynomials in point do so explicitly.
Thus, in conclusion, from the options provided,
None of the given polynomial options accurately represent the polynomial with roots [tex]\( 8 \)[/tex] and [tex]\( 4 \)[/tex].
