Answer :
Sure, let's go through the process of factoring each of these expressions step by step.
35. [tex]\( x^3 + 3x^2 - 10x - 24 \)[/tex]:
1. Use the Rational Root Theorem: The possible rational roots are factors of the constant term, -24. They are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24.
2. Test these values in the polynomial to find a root. Testing, you'll find [tex]\( x = 3 \)[/tex] is a root.
3. Use synthetic division with the root [tex]\( x = 3 \)[/tex] to divide the polynomial by [tex]\( x - 3 \)[/tex].
4. The quotient is a quadratic: [tex]\( x^2 + 6x + 8 \)[/tex].
5. Factor the quadratic: [tex]\( x^2 + 6x + 8 = (x + 2)(x + 4) \)[/tex].
6. Combine all factors: [tex]\( (x - 3)(x + 2)(x + 4) \)[/tex].
36. [tex]\( x^3 + 2x^2 - 13x + 10 \)[/tex]:
1. Use the Rational Root Theorem: The possible rational roots are factors of 10: ±1, ±2, ±5, ±10.
2. Testing these values, you'll find [tex]\( x = 2 \)[/tex] is a root.
3. Synthetic division with [tex]\( x = 2 \)[/tex] gives you the quotient: [tex]\( x^2 + 4x - 5 \)[/tex].
4. Factor the quadratic: [tex]\( x^2 + 4x - 5 = (x - 1)(x + 5) \)[/tex].
5. Factor completely: [tex]\( (x - 2)(x - 1)(x + 5) \)[/tex].
37. [tex]\( 2x^4 - 7x^3 - 23x^2 + 43x - 15 \)[/tex]:
1. Use the Rational Root Theorem: Check possible roots for integer values. You'll find that [tex]\( x = 1 \)[/tex] is a root.
2. Synthetic division by [tex]\( x - 1 \)[/tex] yields [tex]\( 2x^3 - 5x^2 - 28x + 15 \)[/tex].
3. Find another root, say [tex]\( x = 5 \)[/tex] is also a root.
4. Synthetic division again, which yields [tex]\( 2x^2 + 3x - 3 \)[/tex].
5. Factor the quadratic: [tex]\( 2x^2 + 3x - 3 = (x + 3)(2x - 1) \)[/tex].
6. Combine factors: [tex]\( (x - 1)(x - 5)(x + 3)(2x - 1) \)[/tex].
38. [tex]\( 3x^4 - x^3 - 21x^2 - 11x + 6 \)[/tex]:
1. Testing roots, you'll find [tex]\( x = 3 \)[/tex] and [tex]\( x = -1 \)[/tex] are roots.
2. Perform synthetic division for each root to break down the polynomial successively.
3. After dividing, you end up with a quadratic: [tex]\( x^2 + 2x - 2 \)[/tex].
4. Factor the quadratic: [tex]\( x^2 + 2x - 2 = (x + 2)(3x - 1) \)[/tex].
5. Combining all factors: [tex]\( (x - 3)(x + 1)(x + 2)(3x - 1) \)[/tex].
39. [tex]\( 3x^5 - 4x^4 - 23x^3 + 14x^2 + 34x - 12 \)[/tex]:
1. Use the Rational Root Theorem: Find that one root is [tex]\( x = 3 \)[/tex].
2. Use synthetic division: Breaking down gives [tex]\( x^4 - x^3 - 20x^2 + 8x + 12 \)[/tex].
3. Continue looking for roots: [tex]\( x = -2 \)[/tex] works further.
4. After more synthetic division, you're left with a quadratic: [tex]\( x^2 - 2 \)[/tex].
5. This results in the factors: [tex]\( (x^2 - 2)(x + 2)(3x - 1) \)[/tex].
6. Complete factorization: [tex]\( (x - 3)(x + 2)(3x - 1)(x^2 - 2) \)[/tex].
That's how you would factor each expression! If you have any more questions or topics you'd like to discuss, feel free to ask!
35. [tex]\( x^3 + 3x^2 - 10x - 24 \)[/tex]:
1. Use the Rational Root Theorem: The possible rational roots are factors of the constant term, -24. They are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24.
2. Test these values in the polynomial to find a root. Testing, you'll find [tex]\( x = 3 \)[/tex] is a root.
3. Use synthetic division with the root [tex]\( x = 3 \)[/tex] to divide the polynomial by [tex]\( x - 3 \)[/tex].
4. The quotient is a quadratic: [tex]\( x^2 + 6x + 8 \)[/tex].
5. Factor the quadratic: [tex]\( x^2 + 6x + 8 = (x + 2)(x + 4) \)[/tex].
6. Combine all factors: [tex]\( (x - 3)(x + 2)(x + 4) \)[/tex].
36. [tex]\( x^3 + 2x^2 - 13x + 10 \)[/tex]:
1. Use the Rational Root Theorem: The possible rational roots are factors of 10: ±1, ±2, ±5, ±10.
2. Testing these values, you'll find [tex]\( x = 2 \)[/tex] is a root.
3. Synthetic division with [tex]\( x = 2 \)[/tex] gives you the quotient: [tex]\( x^2 + 4x - 5 \)[/tex].
4. Factor the quadratic: [tex]\( x^2 + 4x - 5 = (x - 1)(x + 5) \)[/tex].
5. Factor completely: [tex]\( (x - 2)(x - 1)(x + 5) \)[/tex].
37. [tex]\( 2x^4 - 7x^3 - 23x^2 + 43x - 15 \)[/tex]:
1. Use the Rational Root Theorem: Check possible roots for integer values. You'll find that [tex]\( x = 1 \)[/tex] is a root.
2. Synthetic division by [tex]\( x - 1 \)[/tex] yields [tex]\( 2x^3 - 5x^2 - 28x + 15 \)[/tex].
3. Find another root, say [tex]\( x = 5 \)[/tex] is also a root.
4. Synthetic division again, which yields [tex]\( 2x^2 + 3x - 3 \)[/tex].
5. Factor the quadratic: [tex]\( 2x^2 + 3x - 3 = (x + 3)(2x - 1) \)[/tex].
6. Combine factors: [tex]\( (x - 1)(x - 5)(x + 3)(2x - 1) \)[/tex].
38. [tex]\( 3x^4 - x^3 - 21x^2 - 11x + 6 \)[/tex]:
1. Testing roots, you'll find [tex]\( x = 3 \)[/tex] and [tex]\( x = -1 \)[/tex] are roots.
2. Perform synthetic division for each root to break down the polynomial successively.
3. After dividing, you end up with a quadratic: [tex]\( x^2 + 2x - 2 \)[/tex].
4. Factor the quadratic: [tex]\( x^2 + 2x - 2 = (x + 2)(3x - 1) \)[/tex].
5. Combining all factors: [tex]\( (x - 3)(x + 1)(x + 2)(3x - 1) \)[/tex].
39. [tex]\( 3x^5 - 4x^4 - 23x^3 + 14x^2 + 34x - 12 \)[/tex]:
1. Use the Rational Root Theorem: Find that one root is [tex]\( x = 3 \)[/tex].
2. Use synthetic division: Breaking down gives [tex]\( x^4 - x^3 - 20x^2 + 8x + 12 \)[/tex].
3. Continue looking for roots: [tex]\( x = -2 \)[/tex] works further.
4. After more synthetic division, you're left with a quadratic: [tex]\( x^2 - 2 \)[/tex].
5. This results in the factors: [tex]\( (x^2 - 2)(x + 2)(3x - 1) \)[/tex].
6. Complete factorization: [tex]\( (x - 3)(x + 2)(3x - 1)(x^2 - 2) \)[/tex].
That's how you would factor each expression! If you have any more questions or topics you'd like to discuss, feel free to ask!
