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Choose whole numbers to fill in the two blanks in the following polynomial function:

h(x) = -3x^2(5x +__) (x -___)

3 answers

To fill in the blanks in the polynomial function \( h(x) = -3x^2(5x + __) (x - __) \), we can choose numbers that will maintain the polynomial's structure and form.

Let's analyze the structure:

  1. The term \( 5x + __ \): We need to choose a whole number that will be added to \( 5x \).
  2. The term \( x - __ \): We need to choose a whole number such that when this term is multiplied by \( 5x + __ \), it results in an appropriately factored polynomial.

Example Choices

Let's say we choose \( 2 \) for the first blank and \( 3 \) for the second blank. Thus, we have:

  • \( 5x + 2 \)
  • \( x - 3 \)

Now our polynomial function will look like this:

\[ h(x) = -3x^2(5x + 2)(x - 3) \]

Verification of Choices

You could expand it further if you want to verify:

  1. First multiply the binomials: \[ (5x + 2)(x - 3) = 5x^2 - 15x + 2x - 6 = 5x^2 - 13x - 6 \]

  2. Then multiply by \(-3x^2\): \[ h(x) = -3x^2(5x^2 - 13x - 6) = -15x^4 + 39x^3 + 18x^2 \]

So, the choice works correctly. Thus, one set of results for the blanks is:

  • First blank: \( 2 \)
  • Second blank: \( 3 \)

You could choose other whole numbers as long as they satisfy your requirements.

Find the zeros of h(x)

To find the zeros of the polynomial function

\[ h(x) = -3x^2(5x + 2)(x - 3), \]

we need to set \( h(x) = 0 \) and solve for \( x \).

Step 1: Set the function to zero

\[ -3x^2(5x + 2)(x - 3) = 0. \]

Step 2: Use the zero product property

The zero product property tells us that if a product of multiple factors equals zero, at least one of the factors must be equal to zero. So, we can set each factor to zero:

  1. \( -3x^2 = 0 \)
  2. \( 5x + 2 = 0 \)
  3. \( x - 3 = 0 \)

Step 3: Solve each equation

  1. For \( -3x^2 = 0 \): \[ x^2 = 0 \quad \Rightarrow \quad x = 0. \]

  2. For \( 5x + 2 = 0 \): \[ 5x = -2 \quad \Rightarrow \quad x = -\frac{2}{5}. \]

  3. For \( x - 3 = 0 \): \[ x = 3. \]

Step 4: List the zeros

The zeros of the function \( h(x) \) are:

  • \( x = 0 \)
  • \( x = -\frac{2}{5} \)
  • \( x = 3 \)

Thus, the zeros of \( h(x) \) are \({0, -\frac{2}{5}, 3}\).