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Question

15x^2-10x-40
Factor the trinomial, 7y^4-29y^3-6y^2
5y+31y+6
12x^2+10x+2
2x^2+2x-40
24x^2-28x-12
3x^2+23x-36
2x^2+13x-24
Write the perimeter of the triangle as a simplified polynomial. Then, factor the polynomial of 4x^2-1, 2x+2, and x^2+14x Find the perimeter of the triangle and factor the polynomial.

Answer 1

Let's solve each of your questions one by one.

1. Factor the trinomial: \(15x^2 - 10x - 40\)

To factor \(15x^2 - 10x - 40\):

1. Factor out the greatest common factor (GCF): \[ 5(3x^2 - 2x - 8) \]

2. Now, we can factor the trinomial \(3x^2 - 2x - 8\). We need two numbers that multiply to \(3 \times -8 = -24\) and add to \(-2\). These numbers are \(4\) and \(-6\).

3. Rewrite \(3x^2 - 2x - 8\): \[ 3x^2 + 4x - 6x - 8 \]

4. Factor by grouping: \[ x(3x + 4) - 2(3x + 4) = (3x + 4)(x - 2) \]

5. So, the complete factorization is: \[ 5(3x + 4)(x - 2) \]

2. Factor the trinomial: \(7y^4 - 29y^3 - 6y^2\)

Factor out the GCF first: \[ y^2(7y^2 - 29y - 6) \]

Now, we need to factor \(7y^2 - 29y - 6\). We need numbers that multiply to \(7 \times -6 = -42\) and add to \(-29\). These numbers are \(-30\) and \(1\).

1. Rewrite: \[ 7y^2 - 30y + y - 6 \]

2. Factor by grouping: \[ y(7y - 30) + 1(7y - 6) = (7y - 30)(y + 1) \]

So, the complete factorization is: \[ y^2(7y - 30)(y + 1) \]

3. Simplify the expression: \(5y + 31y + 6\)

Combine like terms: \[ (5y + 31y) + 6 = 36y + 6 \]

4. Factor the trinomial: \(12x^2 + 10x + 2\)

First, factor out the GCF: \[ 2(6x^2 + 5x + 1) \]

Now, we need two numbers that multiply to \(6 \times 1 = 6\) and add to \(5\) (which are \(3\) and \(2\)).

1. Rewrite: \[ 2(6x^2 + 3x + 2x + 1) \]

2. Factor by grouping: \[ 2[(3x + 1)(2x + 1)] \]

So, the complete factorization is: \[ 2(3x + 1)(2x + 1) \]

5. Factor the polynomial: \(2x^2 + 2x - 40\)

Factor out the GCF: \[ 2(x^2 + x - 20) \]

Now, we need two numbers that multiply to \(-20\) and add to \(1\) (which are \(5\) and \(-4\)).

1. Rewrite: \[ 2(x^2 + 5x - 4x - 20) \]

2. Factor by grouping: \[ 2[(x + 5)(x - 4)] \]

So, the complete factorization is: \[ 2(x + 5)(x - 4) \]

6. Factor the polynomial: \(24x^2 - 28x - 12\)

Factor out the GCF: \[ 4(6x^2 - 7x - 3) \]

We need two numbers that multiply to \(-18\) and add to \(-7\) (which are \(-9\) and \(2\)).

1. Rewrite: \[ 4(6x^2 - 9x + 2x - 3) \]

2. Factor by grouping: \[ 4[(3x + 2)(2x - 3)] \]

So, the complete factorization is: \[ 4(3x + 2)(2x - 3) \]

7. Factor the polynomial: \(3x^2 + 23x - 36\)

We need two numbers that multiply to \(-108\) and add to \(23\) (which are \(27\) and \(-4\)).

1. Rewrite: \[ 3x^2 + 27x - 4x - 36 \]

2. Factor by grouping: \[ (3x - 4)(x + 9) \]

8. Factor the polynomial: \(2x^2 + 13x - 24\)

We need two numbers that multiply to \(-48\) and add to \(13\) (which are \(16\) and \(-3\)).

1. Rewrite: \[ 2x^2 + 16x - 3x - 24 \]

2. Factor by grouping: \[ (2x - 3)(x + 8) \]

9: Perimeter of the triangle

Let the sides of the triangle be represented by the polynomials given:

• \(4x^2 - 1\)
• \(2x + 2\)
• \(x^2 + 14x\)

Calculate the perimeter \(P\): \[ P = (4x^2 - 1) + (2x + 2) + (x^2 + 14x) \] Combine like terms: \[ P = 4x^2 + x^2 + 2x + 14x + 2 - 1 = 5x^2 + 16x + 1 \]

Factor the perimeter polynomial: \(5x^2 + 16x + 1\)

We need numbers that multiply to \(5 \times 1 = 5\) and add to \(16\) (which are \(15\) and \(1\)).

1. Rewrite: \[ 5x^2 + 15x + x + 1 \]

2. Factor by grouping: \[ 5x(x + 3) + 1(x + 3) = (5x + 1)(x + 3) \]

Final Results:

1. \(15x^2 - 10x - 40 = 5(3x + 4)(x - 2)\)
2. \(7y^4 - 29y^3 - 6y^2 = y^2(7y - 30)(y + 1)\)
3. Simplified \(5y + 31y + 6 = 36y + 6\)
4. \(12x^2 + 10x + 2 = 2(3x + 1)(2x + 1)\)
5. \(2x^2 + 2x - 40 = 2(x + 5)(x - 4)\)
6. \(24x^2 - 28x - 12 = 4(3x + 2)(2x - 3)\)
7. \(3x^2 + 23x - 36 = (3x - 4)(x + 9)\)
8. \(2x^2 + 13x - 24 = (2x - 3)(x + 8)\)
9. The perimeter \(P = 5x^2 + 16x + 1\) and it factors to \((5x + 1)(x + 3)\).