Explore IDNLearn.com's extensive Q&A database and find the answers you're looking for. Find the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
Certainly! Let's solve each polynomial function step by step to find all other zeros, given one zero for each polynomial.
### 33. [tex]\( f(x) = x^3 - x^2 - 4x - 6 \)[/tex]
Given zero: [tex]\( 3 \)[/tex]
1. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - 3) \cdot Q(x) \)[/tex] where [tex]\( Q(x) \)[/tex] is a quadratic polynomial.
2. Dividing [tex]\( f(x) \)[/tex] by [tex]\( (x - 3) \)[/tex], we can find [tex]\( Q(x) \)[/tex].
3. Solving the quadratic equation [tex]\( Q(x) = 0 \)[/tex] will give the remaining zeros.
The other zeros, found from the calculations, are:
[tex]\[ -1 - i, \quad -1 + i \][/tex]
### 34. [tex]\( f(x) = x^3 + 4x^2 - 5 \)[/tex]
Given zero: [tex]\( 1 \)[/tex]
1. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - 1) \cdot R(x) \)[/tex] where [tex]\( R(x) \)[/tex] is a quadratic polynomial.
2. Dividing [tex]\( f(x) \)[/tex] by [tex]\( (x - 1) \)[/tex], we can find [tex]\( R(x) \)[/tex].
3. Solving the quadratic equation [tex]\( R(x) = 0 \)[/tex] will give the remaining zeros.
The other zeros are:
[tex]\[ -\frac{5}{2} - \frac{\sqrt{5}}{2}, \quad -\frac{5}{2} + \frac{\sqrt{5}}{2} \][/tex]
### 35. [tex]\( f(x) = x^3 - 7x^2 + 17x - 15 \)[/tex]
Given zero: [tex]\( 2 - i \)[/tex]
1. Complex roots come in conjugate pairs, so [tex]\( 2 + i \)[/tex] is also a root.
2. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - (2 - i))(x - (2 + i)) \cdot P(x) \)[/tex] where [tex]\( P(x) \)[/tex] is a linear polynomial.
3. Solving [tex]\( P(x) = 0 \)[/tex] will give the remaining zero.
The calculations lead to these other zeros:
[tex]\[ 3, \quad 2 + i \][/tex]
### 36. [tex]\( f(x) = 4x^3 + 6x^2 - 2x - 1 \)[/tex]
Given zero: [tex]\( \frac{1}{2} \)[/tex]
1. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - \frac{1}{2}) \cdot S(x) \)[/tex] where [tex]\( S(x) \)[/tex] is a quadratic polynomial.
2. Dividing [tex]\( f(x) \)[/tex] by [tex]\( (x - \frac{1}{2}) \)[/tex], we can find [tex]\( S(x) \)[/tex].
3. Solving the quadratic equation [tex]\( S(x) = 0 \)[/tex] will give the remaining zeros.
The other zeros are:
[tex]\[ -1.70710678118655, \quad -0.292893218813452 \][/tex]
### 37. [tex]\( f(x) = x^4 + 5x^2 + 4 \)[/tex]
Given zero: [tex]\( -i \)[/tex]
1. Complex roots come in conjugate pairs, so [tex]\( i \)[/tex] is also a root.
2. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x + i)(x - i) \cdot T(x) \)[/tex] where [tex]\( T(x) \)[/tex] is a quadratic polynomial.
3. Solving [tex]\( T(x) = 0 \)[/tex] will give the remaining zeros.
The calculations result in these other zeros:
[tex]\[ -2i, \quad i, \quad 2i \][/tex]
### 38. [tex]\( f(x) = x^4 + 26x^2 + 25 \)[/tex]
Given zero: [tex]\( i \)[/tex]
1. Complex roots come in conjugate pairs, so [tex]\( -i \)[/tex] is also a root.
2. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - i)(x + i) \cdot U(x) \)[/tex] where [tex]\( U(x) \)[/tex] is a quadratic polynomial.
3. Solving [tex]\( U(x) = 0 \)[/tex] will give the remaining zeros.
The other zeros are:
[tex]\[ -5i, \quad -i, \quad 5i \][/tex]
And there you have it! All other zeros for each given polynomial function.
### 33. [tex]\( f(x) = x^3 - x^2 - 4x - 6 \)[/tex]
Given zero: [tex]\( 3 \)[/tex]
1. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - 3) \cdot Q(x) \)[/tex] where [tex]\( Q(x) \)[/tex] is a quadratic polynomial.
2. Dividing [tex]\( f(x) \)[/tex] by [tex]\( (x - 3) \)[/tex], we can find [tex]\( Q(x) \)[/tex].
3. Solving the quadratic equation [tex]\( Q(x) = 0 \)[/tex] will give the remaining zeros.
The other zeros, found from the calculations, are:
[tex]\[ -1 - i, \quad -1 + i \][/tex]
### 34. [tex]\( f(x) = x^3 + 4x^2 - 5 \)[/tex]
Given zero: [tex]\( 1 \)[/tex]
1. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - 1) \cdot R(x) \)[/tex] where [tex]\( R(x) \)[/tex] is a quadratic polynomial.
2. Dividing [tex]\( f(x) \)[/tex] by [tex]\( (x - 1) \)[/tex], we can find [tex]\( R(x) \)[/tex].
3. Solving the quadratic equation [tex]\( R(x) = 0 \)[/tex] will give the remaining zeros.
The other zeros are:
[tex]\[ -\frac{5}{2} - \frac{\sqrt{5}}{2}, \quad -\frac{5}{2} + \frac{\sqrt{5}}{2} \][/tex]
### 35. [tex]\( f(x) = x^3 - 7x^2 + 17x - 15 \)[/tex]
Given zero: [tex]\( 2 - i \)[/tex]
1. Complex roots come in conjugate pairs, so [tex]\( 2 + i \)[/tex] is also a root.
2. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - (2 - i))(x - (2 + i)) \cdot P(x) \)[/tex] where [tex]\( P(x) \)[/tex] is a linear polynomial.
3. Solving [tex]\( P(x) = 0 \)[/tex] will give the remaining zero.
The calculations lead to these other zeros:
[tex]\[ 3, \quad 2 + i \][/tex]
### 36. [tex]\( f(x) = 4x^3 + 6x^2 - 2x - 1 \)[/tex]
Given zero: [tex]\( \frac{1}{2} \)[/tex]
1. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - \frac{1}{2}) \cdot S(x) \)[/tex] where [tex]\( S(x) \)[/tex] is a quadratic polynomial.
2. Dividing [tex]\( f(x) \)[/tex] by [tex]\( (x - \frac{1}{2}) \)[/tex], we can find [tex]\( S(x) \)[/tex].
3. Solving the quadratic equation [tex]\( S(x) = 0 \)[/tex] will give the remaining zeros.
The other zeros are:
[tex]\[ -1.70710678118655, \quad -0.292893218813452 \][/tex]
### 37. [tex]\( f(x) = x^4 + 5x^2 + 4 \)[/tex]
Given zero: [tex]\( -i \)[/tex]
1. Complex roots come in conjugate pairs, so [tex]\( i \)[/tex] is also a root.
2. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x + i)(x - i) \cdot T(x) \)[/tex] where [tex]\( T(x) \)[/tex] is a quadratic polynomial.
3. Solving [tex]\( T(x) = 0 \)[/tex] will give the remaining zeros.
The calculations result in these other zeros:
[tex]\[ -2i, \quad i, \quad 2i \][/tex]
### 38. [tex]\( f(x) = x^4 + 26x^2 + 25 \)[/tex]
Given zero: [tex]\( i \)[/tex]
1. Complex roots come in conjugate pairs, so [tex]\( -i \)[/tex] is also a root.
2. We can factor [tex]\( f(x) \)[/tex] as [tex]\( (x - i)(x + i) \cdot U(x) \)[/tex] where [tex]\( U(x) \)[/tex] is a quadratic polynomial.
3. Solving [tex]\( U(x) = 0 \)[/tex] will give the remaining zeros.
The other zeros are:
[tex]\[ -5i, \quad -i, \quad 5i \][/tex]
And there you have it! All other zeros for each given polynomial function.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for visiting IDNLearn.com. We’re here to provide accurate and reliable answers, so visit us again soon.
