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Answer :
Sure! Let's solve each part of the problem step-by-step.
### Remainder Theorem
The remainder theorem states that if a polynomial [tex]\( P(x) \)[/tex] is divided by a binomial [tex]\( x-a \)[/tex], the remainder is [tex]\( P(a) \)[/tex].
1. [tex]\( x-1 \)[/tex]
- Find [tex]\( P(1) \)[/tex] for [tex]\( P(x) = 3x^3 + 7x^2 - 2x + 6 \)[/tex].
- [tex]\( P(1) = 3(1)^3 + 7(1)^2 - 2(1) + 6 = 3 + 7 - 2 + 6 = 14 \)[/tex]
- Remainder: 14
2. [tex]\( x-3 \)[/tex]
- Find [tex]\( P(3) \)[/tex].
- [tex]\( P(3) = 3(3)^3 + 7(3)^2 - 2(3) + 6 = 81 + 63 - 6 + 6 = 144 \)[/tex]
- Remainder: 144
3. [tex]\( x+2 \)[/tex]
- Find [tex]\( P(-2) \)[/tex].
- [tex]\( P(-2) = 3(-2)^3 + 7(-2)^2 - 2(-2) + 6 = -24 + 28 + 4 + 6 = 14 \)[/tex]
- Remainder: 14
4. [tex]\( x+1 \)[/tex]
- Find [tex]\( P(-1) \)[/tex].
- [tex]\( P(-1) = 3(-1)^3 + 7(-1)^2 - 2(-1) + 6 = -3 + 7 + 2 + 6 = 12 \)[/tex]
- Remainder: 12
### Factor Theorem
The factor theorem is used to establish if [tex]\( x-a \)[/tex] is a factor of [tex]\( P(x) \)[/tex]. If [tex]\( P(a) = 0 \)[/tex], then [tex]\( x-a \)[/tex] is a factor.
5. [tex]\( x-2 \)[/tex]
- For [tex]\( P(x) = x^4 - 10x^3 + 17x^2 + 52x - 60 \)[/tex], find [tex]\( P(2) \)[/tex].
- [tex]\( P(2) = (2)^4 - 10(2)^3 + 17(2)^2 + 52(2) - 60 = 16 - 80 + 68 + 104 - 60 = 48 \)[/tex]
- Not a factor (since [tex]\( P(2) \neq 0 \)[/tex])
6. [tex]\( x-1 \)[/tex]
- Find [tex]\( P(1) \)[/tex].
- [tex]\( P(1) = 1 - 10 + 17 + 52 - 60 = 0 \)[/tex]
- Is a factor (since [tex]\( P(1) = 0 \)[/tex])
7. [tex]\( x+2 \)[/tex]
- Find [tex]\( P(-2) \)[/tex].
- [tex]\( P(-2) = (-2)^4 - 10(-2)^3 + 17(-2)^2 + 52(-2) - 60 = 16 + 80 + 68 - 104 - 60 = 0 \)[/tex]
- Is a factor (since [tex]\( P(-2) = 0 \)[/tex])
8. [tex]\( x+3 \)[/tex]
- Find [tex]\( P(-3) \)[/tex].
- [tex]\( P(-3) = (-3)^4 - 10(-3)^3 + 17(-3)^2 + 52(-3) - 60 = 81 + 270 + 153 - 156 - 60 = 288 \)[/tex]
- Not a factor (since [tex]\( P(-3) \neq 0 \)[/tex])
### Rational Root Theorem
The rational root theorem states that any possible rational root of a polynomial is in the form [tex]\( \pm \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term, and [tex]\( q \)[/tex] is a factor of the leading coefficient.
9. [tex]\( P(x) = x^3 + 5x^2 - 2x + 6 \)[/tex]
- Factors of the constant term (6): [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex]
- Factors of the leading coefficient (1): [tex]\( \pm 1 \)[/tex]
- Possible rational zeros: [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex]
10. [tex]\( P(x) = x^5 - 8x^4 + 2x^3 + x^2 - 7x - 12 \)[/tex]
- Factors of the constant term (-12): [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex]
- Factors of the leading coefficient (1): [tex]\( \pm 1 \)[/tex]
- Possible rational zeros: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex]
11. [tex]\( P(x) = x^3 + 8 \)[/tex]
- Factors of the constant term (8): [tex]\( \pm 1, \pm 2, \pm 4, \pm 8 \)[/tex]
- Factors of the leading coefficient (1): [tex]\( \pm 1 \)[/tex]
- Possible rational zeros: [tex]\( \pm 1, \pm 2, \pm 4, \pm 8 \)[/tex]
12. [tex]\( P(x) = 4x^3 - 5x^2 + 3x + 7 \)[/tex]
- Factors of the constant term (7): [tex]\( \pm 1, \pm 7 \)[/tex]
- Factors of the leading coefficient (4): [tex]\( \pm 1, \pm 2, \pm 4 \)[/tex]
- Possible rational zeros: [tex]\( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{4} \)[/tex]
I hope this helps! Let me know if you have any questions.
### Remainder Theorem
The remainder theorem states that if a polynomial [tex]\( P(x) \)[/tex] is divided by a binomial [tex]\( x-a \)[/tex], the remainder is [tex]\( P(a) \)[/tex].
1. [tex]\( x-1 \)[/tex]
- Find [tex]\( P(1) \)[/tex] for [tex]\( P(x) = 3x^3 + 7x^2 - 2x + 6 \)[/tex].
- [tex]\( P(1) = 3(1)^3 + 7(1)^2 - 2(1) + 6 = 3 + 7 - 2 + 6 = 14 \)[/tex]
- Remainder: 14
2. [tex]\( x-3 \)[/tex]
- Find [tex]\( P(3) \)[/tex].
- [tex]\( P(3) = 3(3)^3 + 7(3)^2 - 2(3) + 6 = 81 + 63 - 6 + 6 = 144 \)[/tex]
- Remainder: 144
3. [tex]\( x+2 \)[/tex]
- Find [tex]\( P(-2) \)[/tex].
- [tex]\( P(-2) = 3(-2)^3 + 7(-2)^2 - 2(-2) + 6 = -24 + 28 + 4 + 6 = 14 \)[/tex]
- Remainder: 14
4. [tex]\( x+1 \)[/tex]
- Find [tex]\( P(-1) \)[/tex].
- [tex]\( P(-1) = 3(-1)^3 + 7(-1)^2 - 2(-1) + 6 = -3 + 7 + 2 + 6 = 12 \)[/tex]
- Remainder: 12
### Factor Theorem
The factor theorem is used to establish if [tex]\( x-a \)[/tex] is a factor of [tex]\( P(x) \)[/tex]. If [tex]\( P(a) = 0 \)[/tex], then [tex]\( x-a \)[/tex] is a factor.
5. [tex]\( x-2 \)[/tex]
- For [tex]\( P(x) = x^4 - 10x^3 + 17x^2 + 52x - 60 \)[/tex], find [tex]\( P(2) \)[/tex].
- [tex]\( P(2) = (2)^4 - 10(2)^3 + 17(2)^2 + 52(2) - 60 = 16 - 80 + 68 + 104 - 60 = 48 \)[/tex]
- Not a factor (since [tex]\( P(2) \neq 0 \)[/tex])
6. [tex]\( x-1 \)[/tex]
- Find [tex]\( P(1) \)[/tex].
- [tex]\( P(1) = 1 - 10 + 17 + 52 - 60 = 0 \)[/tex]
- Is a factor (since [tex]\( P(1) = 0 \)[/tex])
7. [tex]\( x+2 \)[/tex]
- Find [tex]\( P(-2) \)[/tex].
- [tex]\( P(-2) = (-2)^4 - 10(-2)^3 + 17(-2)^2 + 52(-2) - 60 = 16 + 80 + 68 - 104 - 60 = 0 \)[/tex]
- Is a factor (since [tex]\( P(-2) = 0 \)[/tex])
8. [tex]\( x+3 \)[/tex]
- Find [tex]\( P(-3) \)[/tex].
- [tex]\( P(-3) = (-3)^4 - 10(-3)^3 + 17(-3)^2 + 52(-3) - 60 = 81 + 270 + 153 - 156 - 60 = 288 \)[/tex]
- Not a factor (since [tex]\( P(-3) \neq 0 \)[/tex])
### Rational Root Theorem
The rational root theorem states that any possible rational root of a polynomial is in the form [tex]\( \pm \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term, and [tex]\( q \)[/tex] is a factor of the leading coefficient.
9. [tex]\( P(x) = x^3 + 5x^2 - 2x + 6 \)[/tex]
- Factors of the constant term (6): [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex]
- Factors of the leading coefficient (1): [tex]\( \pm 1 \)[/tex]
- Possible rational zeros: [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex]
10. [tex]\( P(x) = x^5 - 8x^4 + 2x^3 + x^2 - 7x - 12 \)[/tex]
- Factors of the constant term (-12): [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex]
- Factors of the leading coefficient (1): [tex]\( \pm 1 \)[/tex]
- Possible rational zeros: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex]
11. [tex]\( P(x) = x^3 + 8 \)[/tex]
- Factors of the constant term (8): [tex]\( \pm 1, \pm 2, \pm 4, \pm 8 \)[/tex]
- Factors of the leading coefficient (1): [tex]\( \pm 1 \)[/tex]
- Possible rational zeros: [tex]\( \pm 1, \pm 2, \pm 4, \pm 8 \)[/tex]
12. [tex]\( P(x) = 4x^3 - 5x^2 + 3x + 7 \)[/tex]
- Factors of the constant term (7): [tex]\( \pm 1, \pm 7 \)[/tex]
- Factors of the leading coefficient (4): [tex]\( \pm 1, \pm 2, \pm 4 \)[/tex]
- Possible rational zeros: [tex]\( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{4} \)[/tex]
I hope this helps! Let me know if you have any questions.
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