Question

algebra 1
5.11 quiz: using methods of factoring
factor completely.

1. $4x^2 + 8xy + 4y^2$
2. $x^4 + 6x^3 + 9x^2$
3. $5a^2x - 5b^2x$
4. $x^3 + x^2 - x - 1$
5. $a^4 - 16a^2$

Explanation:

Problem 1: \(4x^2 + 8xy + 4y^2\)

Step1: Factor out the GCF

The greatest common factor (GCF) of \(4x^2\), \(8xy\), and \(4y^2\) is \(4\). So we factor out \(4\):
\(4(x^2 + 2xy + y^2)\)

Step2: Factor the perfect square trinomial

The expression \(x^2 + 2xy + y^2\) is a perfect square trinomial, which factors as \((x + y)^2\) (since \((a + b)^2 = a^2 + 2ab + b^2\), here \(a = x\), \(b = y\)).

So putting it together: \(4(x + y)^2\)

Problem 2: \(x^4 + 6x^3 + 9x^2\)

Step1: Factor out the GCF

The GCF of \(x^4\), \(6x^3\), and \(9x^2\) is \(x^2\). Factor out \(x^2\):
\(x^2(x^2 + 6x + 9)\)

Step2: Factor the perfect square trinomial

The expression \(x^2 + 6x + 9\) is a perfect square trinomial, which factors as \((x + 3)^2\) (since \((a + b)^2 = a^2 + 2ab + b^2\), here \(a = x\), \(b = 3\) because \(2 \cdot x \cdot 3 = 6x\) and \(3^2 = 9\)).

So the factored form is: \(x^2(x + 3)^2\)

Problem 3: \(5a^2x - 5b^2x\)

Step1: Factor out the GCF

The GCF of \(5a^2x\) and \(5b^2x\) is \(5x\). Factor out \(5x\):
\(5x(a^2 - b^2)\)

Step2: Factor the difference of squares

The expression \(a^2 - b^2\) is a difference of squares, which factors as \((a - b)(a + b)\) (since \(a^2 - b^2 = (a - b)(a + b)\)).

So the factored form is: \(5x(a - b)(a + b)\)

Problem 4: \(x^3 + x^2 - x - 1\)

Answer:

s:

1. \(4(x + y)^2\)
2. \(x^2(x + 3)^2\)
3. \(5x(a - b)(a + b)\)
4. \((x + 1)^2(x - 1)\)
5. \(a^2(a - 4)(a + 4)\)