Thank you for visiting In Exercises tex 35 40 tex factor the expression into linear factors 35 tex x 3 3x 2 10x 24 tex 36 tex x 3. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
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!
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