Answer :
Sure! Let's go through each polynomial and find its linear factors and zeros.
### 59. [tex]\( f(x) = x^2 + 36 \)[/tex]
1. This is a quadratic equation in the form of [tex]\( x^2 + a^2 \)[/tex].
2. It can be factored as [tex]\( (x - 6i)(x + 6i) \)[/tex].
3. The zeros are [tex]\( x = -6i \)[/tex] and [tex]\( x = 6i \)[/tex].
### 60. [tex]\( f(x) = x^2 + 49 \)[/tex]
1. This is a quadratic equation in the form of [tex]\( x^2 + a^2 \)[/tex].
2. It can be factored as [tex]\( (x - 7i)(x + 7i) \)[/tex].
3. The zeros are [tex]\( x = -7i \)[/tex] and [tex]\( x = 7i \)[/tex].
### 61. [tex]\( h(x) = x^2 - 2x + 17 \)[/tex]
1. This is a standard quadratic equation.
2. It cannot be factored further into real numbers, but can be expressed using complete the square or use the quadratic formula.
3. Zeros are complex: [tex]\( x = 1 - 4i \)[/tex] and [tex]\( x = 1 + 4i \)[/tex].
### 62. [tex]\( g(x) = x^2 + 10x + 17 \)[/tex]
1. A quadratic equation that does not factor into real numbers.
2. Use the quadratic formula to find zeros:
- Zeros: [tex]\( x = -5 - 2\sqrt{2} \)[/tex] and [tex]\( x = -5 + 2\sqrt{2} \)[/tex].
### 63. [tex]\( f(x) = x^4 - 16 \)[/tex]
1. This is a difference of squares: [tex]\( (x^2)^2 - (4)^2 \)[/tex].
2. It factors to [tex]\( (x^2 - 4)(x^2 + 4) \)[/tex].
3. Further factor [tex]\( x^2 - 4 \)[/tex] to [tex]\( (x - 2)(x + 2) \)[/tex].
4. The zeros are [tex]\( x = -2, 2, -2i, 2i \)[/tex].
### 64. [tex]\( f(y) = y^4 - 256 \)[/tex]
1. This is a difference of squares: [tex]\( (y^2)^2 - (16)^2 \)[/tex].
2. It factors to [tex]\( (y^2 - 16)(y^2 + 16) \)[/tex].
3. Further factor [tex]\( y^2 - 16 \)[/tex] to [tex]\( (y - 4)(y + 4) \)[/tex].
4. The zeros are [tex]\( y = -4, 4 \)[/tex].
### 65. [tex]\( f(z) = z^2 - 2z + 2 \)[/tex]
1. This is a standard quadratic equation.
2. Cannot be factored using integers but find zeros using quadratic formula:
- The zeros are complex.
### 66. [tex]\( h(x) = x^3 - 3x^2 + 4x - 2 \)[/tex]
1. Polynomial of degree 3.
2. Factor to get [tex]\( (x - 1)(x^2 - 2x + 2) \)[/tex].
3. The zeros are [tex]\( x = 1, 1 - i, 1 + i \)[/tex].
### 67. [tex]\( g(x) = x^3 - 3x^2 + x + 5 \)[/tex]
1. Polynomial of degree 3.
2. Factor to [tex]\( (x + 1)(x^2 - 4x + 5) \)[/tex].
3. The zeros are [tex]\( x = -1, 2 - i, 2 + i \)[/tex].
### 68. [tex]\( f(x) = x^3 - x^2 + x + 39 \)[/tex]
1. Polynomial of degree 3.
2. Factor to [tex]\( (x + 3)(x^2 - 4x + 13) \)[/tex].
3. The zeros are [tex]\( x = -3, 2 - 3i, 2 + 3i \)[/tex].
### 69. [tex]\( g(x) = x^4 - 4x^3 + 8x^2 - 16x + 16 \)[/tex]
1. This is a polynomial of degree 4.
2. Factor to [tex]\( (x - 2)^2(x^2 + 4) \)[/tex].
3. The zeros are [tex]\( x = 2, -2i, 2i \)[/tex].
### 70. [tex]\( h(x) = x^4 + 6x^3 + 10x^2 + 6x + 9 \)[/tex]
1. This is a polynomial of degree 4.
2. Factor to [tex]\( (x + 3)^2(x^2 + 1) \)[/tex].
3. The zeros are [tex]\( x = -3, -i, i \)[/tex].
These are the factors and zeros for each polynomial in the given exercises.
### 59. [tex]\( f(x) = x^2 + 36 \)[/tex]
1. This is a quadratic equation in the form of [tex]\( x^2 + a^2 \)[/tex].
2. It can be factored as [tex]\( (x - 6i)(x + 6i) \)[/tex].
3. The zeros are [tex]\( x = -6i \)[/tex] and [tex]\( x = 6i \)[/tex].
### 60. [tex]\( f(x) = x^2 + 49 \)[/tex]
1. This is a quadratic equation in the form of [tex]\( x^2 + a^2 \)[/tex].
2. It can be factored as [tex]\( (x - 7i)(x + 7i) \)[/tex].
3. The zeros are [tex]\( x = -7i \)[/tex] and [tex]\( x = 7i \)[/tex].
### 61. [tex]\( h(x) = x^2 - 2x + 17 \)[/tex]
1. This is a standard quadratic equation.
2. It cannot be factored further into real numbers, but can be expressed using complete the square or use the quadratic formula.
3. Zeros are complex: [tex]\( x = 1 - 4i \)[/tex] and [tex]\( x = 1 + 4i \)[/tex].
### 62. [tex]\( g(x) = x^2 + 10x + 17 \)[/tex]
1. A quadratic equation that does not factor into real numbers.
2. Use the quadratic formula to find zeros:
- Zeros: [tex]\( x = -5 - 2\sqrt{2} \)[/tex] and [tex]\( x = -5 + 2\sqrt{2} \)[/tex].
### 63. [tex]\( f(x) = x^4 - 16 \)[/tex]
1. This is a difference of squares: [tex]\( (x^2)^2 - (4)^2 \)[/tex].
2. It factors to [tex]\( (x^2 - 4)(x^2 + 4) \)[/tex].
3. Further factor [tex]\( x^2 - 4 \)[/tex] to [tex]\( (x - 2)(x + 2) \)[/tex].
4. The zeros are [tex]\( x = -2, 2, -2i, 2i \)[/tex].
### 64. [tex]\( f(y) = y^4 - 256 \)[/tex]
1. This is a difference of squares: [tex]\( (y^2)^2 - (16)^2 \)[/tex].
2. It factors to [tex]\( (y^2 - 16)(y^2 + 16) \)[/tex].
3. Further factor [tex]\( y^2 - 16 \)[/tex] to [tex]\( (y - 4)(y + 4) \)[/tex].
4. The zeros are [tex]\( y = -4, 4 \)[/tex].
### 65. [tex]\( f(z) = z^2 - 2z + 2 \)[/tex]
1. This is a standard quadratic equation.
2. Cannot be factored using integers but find zeros using quadratic formula:
- The zeros are complex.
### 66. [tex]\( h(x) = x^3 - 3x^2 + 4x - 2 \)[/tex]
1. Polynomial of degree 3.
2. Factor to get [tex]\( (x - 1)(x^2 - 2x + 2) \)[/tex].
3. The zeros are [tex]\( x = 1, 1 - i, 1 + i \)[/tex].
### 67. [tex]\( g(x) = x^3 - 3x^2 + x + 5 \)[/tex]
1. Polynomial of degree 3.
2. Factor to [tex]\( (x + 1)(x^2 - 4x + 5) \)[/tex].
3. The zeros are [tex]\( x = -1, 2 - i, 2 + i \)[/tex].
### 68. [tex]\( f(x) = x^3 - x^2 + x + 39 \)[/tex]
1. Polynomial of degree 3.
2. Factor to [tex]\( (x + 3)(x^2 - 4x + 13) \)[/tex].
3. The zeros are [tex]\( x = -3, 2 - 3i, 2 + 3i \)[/tex].
### 69. [tex]\( g(x) = x^4 - 4x^3 + 8x^2 - 16x + 16 \)[/tex]
1. This is a polynomial of degree 4.
2. Factor to [tex]\( (x - 2)^2(x^2 + 4) \)[/tex].
3. The zeros are [tex]\( x = 2, -2i, 2i \)[/tex].
### 70. [tex]\( h(x) = x^4 + 6x^3 + 10x^2 + 6x + 9 \)[/tex]
1. This is a polynomial of degree 4.
2. Factor to [tex]\( (x + 3)^2(x^2 + 1) \)[/tex].
3. The zeros are [tex]\( x = -3, -i, i \)[/tex].
These are the factors and zeros for each polynomial in the given exercises.
