Question

1. $3x^2 - 13x + 12$
2. $10x^2 - 42x - 108$
3. $3m^2 + 5m - 12$
4. $7x^2 - 24x + 20$
5. $v^3 - 7v^2 + 6v$
6. $k^3 - 7k^2$
7. $5x^2 - 45x + 100$
8. $v^4 - 11v^3 + 30v^2$

Explanation:

Step1: Factor \(3x^2 - 13x + 12\)

We need two numbers that multiply to \(3\times12 = 36\) and add up to \(-13\). The numbers are \(-9\) and \(-4\).
Rewrite the middle term: \(3x^2 - 9x - 4x + 12\)
Group the terms: \((3x^2 - 9x) + (-4x + 12)\)
Factor out the GCF from each group: \(3x(x - 3) - 4(x - 3)\)
Factor out \((x - 3)\): \((3x - 4)(x - 3)\)

Step2: Factor \(10x^2 - 42x - 108\)

First, factor out the GCF of \(2\): \(2(5x^2 - 21x - 54)\)
Now, factor \(5x^2 - 21x - 54\). We need two numbers that multiply to \(5\times(-54)=-270\) and add up to \(-21\). The numbers are \(-30\) and \(9\).
Rewrite the middle term: \(5x^2 - 30x + 9x - 54\)
Group the terms: \((5x^2 - 30x) + (9x - 54)\)
Factor out the GCF from each group: \(5x(x - 6) + 9(x - 6)\)
Factor out \((x - 6)\): \(2(5x + 9)(x - 6)\)

Step3: Factor \(3m^2 + 5m - 12\)

We need two numbers that multiply to \(3\times(-12)=-36\) and add up to \(5\). The numbers are \(9\) and \(-4\).
Rewrite the middle term: \(3m^2 + 9m - 4m - 12\)
Group the terms: \((3m^2 + 9m) + (-4m - 12)\)
Factor out the GCF from each group: \(3m(m + 3) - 4(m + 3)\)
Factor out \((m + 3)\): \((3m - 4)(m + 3)\)

Step4: Factor \(7x^2 - 24x + 20\)

We need two numbers that multiply to \(7\times20 = 140\) and add up to \(-24\). The numbers are \(-14\) and \(-10\).
Rewrite the middle term: \(7x^2 - 14x - 10x + 20\)
Group the terms: \((7x^2 - 14x) + (-10x + 20)\)
Factor out the GCF from each group: \(7x(x - 2) - 10(x - 2)\)
Factor out \((x - 2)\): \((7x - 10)(x - 2)\)

Step5: Factor \(v^3 - 7v^2 + 6v\)

First, factor out the GCF of \(v\): \(v(v^2 - 7v + 6)\)
Now, factor \(v^2 - 7v + 6\). We need two numbers that multiply to \(6\) and add up to \(-7\). The numbers are \(-6\) and \(-1\).
So, \(v(v - 6)(v - 1)\)

Step6: Factor \(k^3 - 7k^2\)

Factor out the GCF of \(k^2\): \(k^2(k - 7)\)

Step7: Factor \(5x^2 - 45x + 100\)

First, factor out the GCF of \(5\): \(5(x^2 - 9x + 20)\)
Now, factor \(x^2 - 9x + 20\). We need two numbers that multiply to \(20\) and add up to \(-9\). The numbers are \(-5\) and \(-4\).
So, \(5(x - 5)(x - 4)\)

Step8: Factor \(v^4 - 11v^3 + 30v^2\)

First, factor out the GCF of \(v^2\): \(v^2(v^2 - 11v + 30)\)
Now, factor \(v^2 - 11v + 30\). We need two numbers that multiply to \(30\) and add up to \(-11\). The numbers are \(-6\) and \(-5\).
So, \(v^2(v - 6)(v - 5)\)

Answer:

1. \((3x - 4)(x - 3)\)
2. \(2(5x + 9)(x - 6)\)
3. \((3m - 4)(m + 3)\)
4. \((7x - 10)(x - 2)\)
5. \(v(v - 6)(v - 1)\)
6. \(k^2(k - 7)\)
7. \(5(x - 5)(x - 4)\)
8. \(v^2(v - 6)(v - 5)\)