Question

example: write each equation in factored form.
$x^3 + 8x^2 + 16x$
$3x^3 - 3x^2 - 36x$
you try it: what are the roots of the function $ - 4x - 4$? (factor and solve)

Explanation:

For the expression \(x^3 + 8x^2 + 16x\)

Step1: Factor out the GCF

The greatest common factor (GCF) of \(x^3\), \(8x^2\), and \(16x\) is \(x\). So we factor out \(x\):
\(x^3 + 8x^2 + 16x = x(x^2 + 8x + 16)\)

Step2: Factor the quadratic

The quadratic \(x^2 + 8x + 16\) is a perfect square trinomial, since \(8x = 2\times x\times4\) and \(16 = 4^2\). So it factors as \((x + 4)^2\).
Putting it all together, we get:
\(x^3 + 8x^2 + 16x = x(x + 4)^2\)

For the expression \(3x^3 - 3x^2 - 36x\)

Step1: Factor out the GCF

The GCF of \(3x^3\), \(-3x^2\), and \(-36x\) is \(3x\). Factor out \(3x\):
\(3x^3 - 3x^2 - 36x = 3x(x^2 - x - 12)\)

Step2: Factor the quadratic

We need two numbers that multiply to \(-12\) and add to \(-1\). The numbers are \(-4\) and \(3\) (since \(-4\times3=-12\) and \(-4 + 3=-1\)). So we factor the quadratic as \((x - 4)(x + 3)\).
Putting it all together, we get:
\(3x^3 - 3x^2 - 36x = 3x(x - 4)(x + 3)\)

For the "You try it" problem: Find the roots of \(f(x)=x^3 + x^2 - 4x - 4\)

Step1: Factor by grouping

Group the first two terms and the last two terms:
\(f(x)=(x^3 + x^2)+(-4x - 4)\)
Factor out the GCF from each group:
\(f(x)=x^2(x + 1)-4(x + 1)\)
Now, factor out the common binomial factor \((x + 1)\):
\(f(x)=(x + 1)(x^2 - 4)\)

Step2: Factor the difference of squares

The term \(x^2 - 4\) is a difference of squares (\(a^2 - b^2=(a - b)(a + b)\)) with \(a = x\) and \(b = 2\). So we factor it as \((x - 2)(x + 2)\).
Putting it all together, we have:
\(f(x)=(x + 1)(x - 2)(x + 2)\)

Step3: Solve for the roots

To find the roots, we set \(f(x) = 0\):
\((x + 1)(x - 2)(x + 2)=0\)
Using the zero - product property (if \(ab = 0\), then either \(a = 0\) or \(b = 0\)), we get:
\(x+1 = 0\) or \(x - 2 = 0\) or \(x + 2 = 0\)
Solving these equations:
For \(x+1 = 0\), we get \(x=-1\)
For \(x - 2 = 0\), we get \(x = 2\)
For \(x + 2 = 0\), we get \(x=-2\)

Answer:

s:

• Factored form of \(x^3 + 8x^2 + 16x\): \(\boldsymbol{x(x + 4)^2}\)
• Factored form of \(3x^3 - 3x^2 - 36x\): \(\boldsymbol{3x(x - 4)(x + 3)}\)
• Roots of \(x^3 + x^2 - 4x - 4\): \(\boldsymbol{x=-1,x = 2,x=-2}\)