Question

question 4
solve the following polynomial equation.
$2x^4 = 32x^2$

question 5
find the degree 5 polynomial function of the graph below.

Explanation:

Step1: Rearrange the equation

Subtract \(32x^2\) from both sides to get \(2x^4 - 32x^2 = 0\).

Step2: Factor out common terms

Factor out \(2x^2\) from the left - hand side: \(2x^2(x^2 - 16)=0\).

Step3: Factor the difference of squares

We know that \(a^2 - b^2=(a + b)(a - b)\), so \(x^2-16=(x + 4)(x - 4)\). Then the equation becomes \(2x^2(x + 4)(x - 4)=0\).

Step4: Solve for x

Set each factor equal to zero:

• For \(2x^2 = 0\), divide both sides by 2: \(x^2=0\), then \(x = 0\) (with multiplicity 2).
• For \(x + 4=0\), subtract 4 from both sides: \(x=-4\).
• For \(x - 4=0\), add 4 to both sides: \(x = 4\).

Answer:

The solutions of the equation \(2x^4 = 32x^2\) are \(x = 0\), \(x=-4\) and \(x = 4\) (where \(x = 0\) has a multiplicity of 2). In set notation, the solution set is \(\{-4,0,4\}\) (or we can write the solutions as \(x=-4\), \(x = 0\), \(x = 4\)).