Question

mhf4u1 in class assignment – factor theorem
name: ______________
answer all questions in the space provided.

given the following polynomial:
i) ( f(x) = x^5 - 18x^3 + 81x )
ii) ( g(x) = 5x^5 - 12x^4 - 11x^3 + 26x^2 + 12x - 8 )
iii) ( h(x) = 3x^5 - 2x^4 + 24x^3 - 16x^2 - 27x + 18 )

a) using the r.r.t. determine the possible rational roots.
b) determine all the possible root combinations (real or complex).
c) factor fully.
d) sketch the graph.
e) determine the intervals in which the ( f(x) geq 0 )

Explanation:

Step1: Identify the polynomial

We are given \( f(x) = x^5 - 18x^3 + 81x \). First, we can factor out the greatest common factor (GCF) from the polynomial. The GCF of \( x^5 \), \( -18x^3 \), and \( 81x \) is \( x \).
So, \( f(x)=x(x^4 - 18x^2 + 81) \)

Step2: Factor the quartic polynomial

Notice that the quartic polynomial \( x^4 - 18x^2 + 81 \) is a quadratic in terms of \( x^2 \). Let \( y = x^2 \), then the polynomial becomes \( y^2 - 18y + 81 \).
We can factor this quadratic using the formula \( a^2-2ab + b^2=(a - b)^2 \). Here, \( a = y \) and \( b = 9 \) since \( 2ab=2\times y\times9 = 18y \) and \( b^2 = 81 \).
So, \( y^2 - 18y + 81=(y - 9)^2 \)

Step3: Substitute back \( y=x^2 \)

Substituting \( y = x^2 \) back into the factored form, we get \( (x^2 - 9)^2 \)

Step4: Factor the difference of squares

We know that \( a^2 - b^2=(a - b)(a + b) \). For \( x^2 - 9 \), \( a=x \) and \( b = 3 \), so \( x^2 - 9=(x - 3)(x + 3) \)

Step5: Write the fully factored form

Since \( (x^2 - 9)^2=[(x - 3)(x + 3)]^2=(x - 3)^2(x + 3)^2 \)
And we had factored out \( x \) earlier, the fully factored form of \( f(x) \) is:
\( f(x)=x(x - 3)^2(x + 3)^2 \)

Answer:

The fully factored form of \( f(x)=x^5 - 18x^3 + 81x \) is \( \boldsymbol{x(x - 3)^2(x + 3)^2} \)