Question

find the following information pertaining to the polynomial, f(x), graphed above. note: type even or odd for the multiplicities of each zero below. you must have the corresponding multiplicity correct for each zero in order to receive credit! (a) the zeros of f(x) are x = (separate by commas) and the zero x = has multiplicity and, the zero x = has multiplicity. (b) the degree of f(x) is. (c) the leading coefficient of f(x) is. you have 2 attempt(s) remaining before you will receive a new version of this problem.

Explanation:

Step1: Identify Zeros from Graph

Zeros of a polynomial are the x - intercepts (where the graph crosses or touches the x - axis). From the graph, the polynomial touches or crosses the x - axis at \(x=-3\) (assuming the left x - intercept is at \(x = - 3\), maybe the grid was misread earlier, but looking at the graph, the left zero is at \(x=-3\) and the right at \(x = 1\))? Wait, no, looking at the grid, the left x - intercept is at \(x=-3\)? Wait, the x - axis has - 5, and then the intercept is at \(x=-3\)? Wait, no, the graph touches the x - axis at two points: let's see the x - coordinates. The left point is at \(x=-3\)? Wait, no, the grid lines: each square is 1 unit? Let's check the x - axis. The left intercept is at \(x=-3\)? Wait, no, the graph shows the left zero at \(x=-3\) (maybe) and right at \(x = 1\). Wait, actually, looking at the graph, the two x - intercepts are at \(x=-3\) (wait, no, the first intercept (left) is at \(x=-3\)? Wait, no, the x - axis has - 5, then the next grid line is - 4, - 3, - 2, - 1, 0, 1. So the left zero is at \(x=-3\) (where the graph touches the x - axis and turns around) and the right zero is at \(x = 1\) (where the graph touches the x - axis and turns around)? Wait, no, the left part: the graph comes from below, touches the x - axis at \(x=-3\) (so it's a touch, meaning even multiplicity) and then goes up, and the right part touches the x - axis at \(x = 1\) (also a touch, even multiplicity). Wait, but let's re - examine. Wait, the graph: at \(x=-3\), the graph touches the x - axis (so it's a zero with even multiplicity) and at \(x = 1\), the graph touches the x - axis (even multiplicity). Wait, maybe the left zero is \(x=-3\) and right is \(x = 1\).

Step1 (Correct): Identify Zeros

The zeros of the polynomial are the x - values where the graph intersects or touches the x - axis. From the graph, the polynomial touches the x - axis at \(x=-3\) and \(x = 1\).

Step2: Determine Multiplicity

• For a zero, if the graph touches the x - axis (turns around) at that point, the multiplicity is even. If it crosses the x - axis, the multiplicity is odd. Here, at both \(x=-3\) and \(x = 1\), the graph touches the x - axis (does not cross), so both have even multiplicity.

(a) The zeros of \(f(x)\) are \(x=-3,1\). The zero \(x=-3\) has even multiplicity and the zero \(x = 1\) has even multiplicity.

(b) The degree of a polynomial is the sum of the multiplicities of its zeros. Since both zeros have even multiplicity (let's assume multiplicity 2 for each, as the simplest even multiplicity), the degree is \(2 + 2=4\) (a 4th - degree polynomial).

(c) The leading coefficient: To determine the leading coefficient, we look at the end - behavior. As \(x\to\infty\), the graph goes down, and as \(x\to-\infty\), the graph goes down. For a polynomial of even degree, if the leading coefficient is negative, the ends go down. So the leading coefficient is negative.

Part (a)

The zeros of \(f(x)\) are \(x=-3,1\). The zero \(x=-3\) has even multiplicity and the zero \(x = 1\) has even multiplicity.

Part (b)

The degree of \(f(x)\): Since we have two zeros, each with multiplicity 2 (the simplest even multiplicity), the degree is \(2 + 2=4\).

Part (c)

The leading coefficient: As \(x\to\pm\infty\), the graph falls (both ends go down). For a polynomial \(f(x)=a_nx^n+\cdots+a_0\), if \(n\) is even and \(a_n<0\), the ends fall. So the leading coefficient is negative.

Answer:

(a) The zeros of \(f(x)\) are \(x=\boldsymbol{-3,1}\); the zero \(x = \boldsymbol{-3}\) has \(\boldsymbol{even}\) multiplicity and the zero \(x=\boldsymbol{1}\) has \(\boldsymbol{even}\) multiplicity.
(b) The degree of \(f(x)\) is \(\boldsymbol{4}\).
(c) The leading coefficient of \(f(x)\) is \(\boldsymbol{negative}\).