91Ó°ÊÓ

The Rational Zeros Theorem is a tool to find possible rational zeros for a polynomial function. It states that any rational zero of a polynomial equation, formed by coefficient terms, is a fraction where the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient.

To apply this theorem to the polynomial function \(f(x) = x^3 + x^2 - 8x - 12\),

let's list the factors:

• The constant term is -12, so factors are \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12
• The leading coefficient is 1, so factors are \pm 1

Therefore, the possible rational zeros are \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12.

By evaluating each possible rational zero, we can determine which actual zero(s) exist. For instance, substituting \(x = -2\) into the equation results in zero, confirming -2 as a rational zero.

Polynomial division, specifically synthetic division, helps you divide a polynomial by a binomial of the form (x - c).

For our given polynomial function \(f(x) = x^3 + x^2 - 8x - 12\),

After finding that \(x = -2\) is a root using the Rational Zeros Theorem, we can use synthetic division with \(x + 2\).

The synthetic division process resembles long division but is simpler:

• Write down the coefficients: 1, 1, -8, -12
• Bring down the leading coefficient (1)
• Multiply this by -2 and add the result to the next coefficient, repeating the process across all coefficients

After completing the synthetic division for our polynomial:

\ x^3 + x^2 - 8x - 12 \div (x + 2) = x^2 - x - 6\ This quotient polynomial can then be further factored.

Factoring polynomials entails breaking down a complex polynomial into a product of simpler polynomials. For our quotient polynomial \[x^2 - x - 6\]:

• Identify two numbers that multiply to -6 (constant term) and add to -1 (coefficient of the x term)
• In this case, we find -3 and 2

Therefore, \[x^2 - x - 6 = (x - 3)(x + 2)\] Using the factor from the synthetic division, we complete the factorization:

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

This factorized form represents the polynomial function in its simplest form.

Graphing polynomial functions requires understanding the function's zeros and how it behaves around them. For \[f(x) = (x + 2)^2(x - 3)\]:

• Zeros are at \ x = -2 (with multiplicity 2) and x = 3 (with multiplicity 1)

Now, observe the behavior at each zero:

• At \ x = -2 \, the graph touches and rebounds off the x-axis due to multiplicity 2
• At \ x = 3 \, it crosses the x-axis due to multiplicity 1

Be sure to account for end behavior dictated by the polynomial's leading term, \[x^3\]:

• As x approaches \ -\infty \, the graph heads towards \ -\infty \
• As x approaches \ \infty \, the graph heads towards \ \infty

Combine these points to sketch an accurate graph of the polynomial function.