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When solving for the x-intercepts of a polynomial, factoring is often the first step. Factoring involves rewriting the polynomial as a product of simpler polynomials. This makes it easier to find the roots. In the given exercise, the polynomial is: \(f(x) = x^4 - 16\)We notice that \(x^4 - 16\) is a difference of squares. That is because any term of the form \(a^2 - b^2\) can be factored into \((a - b)(a + b)\). Hence:\(x^4 - 16 = (x^2 - 4)(x^2 + 4)\)We continue factoring \(x^2 - 4\), another difference of squares:\(x^2 - 4 = (x - 2)(x + 2)\)By factoring, we obtain:\(x^4 - 16 = (x - 2)(x + 2)(x^2 + 4)\).Note, \(x^2 + 4\) does not factor further in the real number domain, producing complex roots which do not affect x-intercepts.

Once the polynomial is factored, finding the roots (or x-intercepts) becomes straightforward. Set each factor equal to zero and solve for \(x\).From the factors \((x - 2)(x + 2)(x^2 + 4)\), we set each factor to zero:1. \((x - 2) = 0 \rightarrow x = 2\)
2. \((x + 2) = 0 \rightarrow x = -2\)
3. \((x^2 + 4) = 0 \rightarrow x^2 = -4\), and thus \(x = \textpm 2i\).The roots are \(x = 2\) and \(x = -2\). Note, \(x = \textpm 2i\) are complex roots and do not provide real x-intercepts.

The multiplicity of a root determines behavior at that intercept. It is the number of times a root is repeated in the factorization of the polynomial. If a root appears \(n\) times, its multiplicity is \(n\). For our example, each of the real factors, \((x - 2)\) and \((x + 2)\), appears once, indicating a multiplicity of 1 for \(x = 2\) and \(x = -2\).For roots of even multiplicity, the graph touches the x-axis and turns around at the intercept. For roots of odd multiplicity, the graph crosses the x-axis. In this case, the multiplicity of 1 (odd) suggests the graph crosses the x-axis at \(x = 2\) and \(x = -2\).

Understanding graph behavior at intercepts helps visualize polynomial functions. For the polynomial \(f(x) = x^4 - 16\), the x-intercepts are \(x = 2\) and \(x = -2\). At these intercepts, we analyze the graph's behavior:• Since both \(x = 2\) and \(x = -2\) have odd multiplicities (1), the graph crosses the x-axis at these points. However, because the original polynomial is of even degree (4), the overall shape of the graph near these intercepts has a 'touch and turn' behavior—but still considers the overall shape dictated by its leading coefficient.• The polynomial quickly moves upwards on either side of the intercepts due to the \(x^4\) term's dominant role. This results in a sharp U-shape near the x-intercepts, touching but not turning back downward.