Question

factor completely: (6 points each)

1. $3x^3 - 5x^2 + 2x$
2. $28x^4 + 16x^3 - 80x^2$
3. $30x^2y - 87xy + 30y$

Explanation:

Problem 7: \(3x^3 - 5x^2 + 2x\)

Step 1: Factor out the GCF

The greatest common factor (GCF) of \(3x^3\), \(-5x^2\), and \(2x\) is \(x\).
Factor out \(x\):
\(x(3x^2 - 5x + 2)\)

Step 2: Factor the quadratic

We need two numbers that multiply to \(3 \times 2 = 6\) and add to \(-5\). The numbers are \(-2\) and \(-3\).
Rewrite the middle term:
\(3x^2 - 3x - 2x + 2\)

Step 3: Group and factor

Group the first two and last two terms:
\((3x^2 - 3x) + (-2x + 2)\)
Factor out GCF from each group:
\(3x(x - 1) - 2(x - 1)\)

Step 4: Factor out the common binomial

Factor out \((x - 1)\):
\(x(3x - 2)(x - 1)\)

Problem 8: \(28x^4 + 16x^3 - 80x^2\)

Step 1: Factor out the GCF

The GCF of \(28x^4\), \(16x^3\), and \(-80x^2\) is \(4x^2\).
Factor out \(4x^2\):
\(4x^2(7x^2 + 4x - 20)\)

Step 2: Factor the quadratic

We need two numbers that multiply to \(7 \times (-20) = -140\) and add to \(4\). The numbers are \(14\) and \(-10\).
Rewrite the middle term:
\(7x^2 + 14x - 10x - 20\)

Step 3: Group and factor

Group the first two and last two terms:
\((7x^2 + 14x) + (-10x - 20)\)
Factor out GCF from each group:
\(7x(x + 2) - 10(x + 2)\)

Step 4: Factor out the common binomial

Factor out \((x + 2)\):
\(4x^2(7x - 10)(x + 2)\)

Problem 9: \(30x^2y - 87xy + 30y\)

Step 1: Factor out the GCF

The GCF of \(30x^2y\), \(-87xy\), and \(30y\) is \(3y\).
Factor out \(3y\):
\(3y(10x^2 - 29x + 10)\)

Step 2: Factor the quadratic

We need two numbers that multiply to \(10 \times 10 = 100\) and add to \(-29\). The numbers are \(-25\) and \(-4\).
Rewrite the middle term:
\(10x^2 - 25x - 4x + 10\)

Step 3: Group and factor

Group the first two and last two terms:
\((10x^2 - 25x) + (-4x + 10)\)
Factor out GCF from each group:
\(5x(2x - 5) - 2(2x - 5)\)

Step 4: Factor out the common binomial

Factor out \((2x - 5)\):
\(3y(5x - 2)(2x - 5)\)

Answer:

s:

1. \(\boldsymbol{x(3x - 2)(x - 1)}\)
2. \(\boldsymbol{4x^2(7x - 10)(x + 2)}\)
3. \(\boldsymbol{3y(5x - 2)(2x - 5)}\)