We appreciate your visit to 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 offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
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!
Thanks for taking the time to read 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. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
