A)Write the product as linear factors: (x2−4)(x2−9).(1 point)

Question

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Question 2
A)Decompose the polynomial x3−2x2−5x+6 into its linear factors, given that x−1 is a factor. (1 point)
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Question 3
A)How many x-intercepts does the function f(x)=3(2x−1)(x+2)(8x−5)2 have when graphed? (1 point)
The function f(x) has $$ x-intercepts when graphed.
Question 4
A)
Use the image to answer the question.

A curve with 3 inflection points is plotted on a coordinate grid. The plot starts in quadrant 2, descends into quadrant 3, rises to a turning point on the x-axis left of the origin, descends to a turning point in quadrant 4 and rises through quadrant 1.

Find the zeros of the polynomial function j(x)=64x4+160x3+48x2−40x−16, given its graph. Enter the zeros in order from least to greatest. Enter any non-integer numbers in decimal form.

(2 points)
The zeros of j(x) are x=, x=, and x=
Question 5
A)
Use the image to answer the question.

A curve with 2 turning points is plotted on a grid and is labeled q left parenthesis x right parenthesis. The plot starts in quadrant 3, rises to a turning point in quadrant 2, drops to a turning point in quadrant 3, and rises through quadrants 4 and 1.

Find the missing values in the factored form of q(x), given its graph.

(1 point)
q(x)=(x+4)(x+ )(x− )​​

Answer 1

Here are the solutions to the questions you've provided:

Question 1:

A) To write the product \((x^2 - 4)(x^2 - 9)\) as linear factors, we can factor each term:

• \(x^2 - 4\) is a difference of squares: \[ x^2 - 4 = (x - 2)(x + 2) \]
• \(x^2 - 9\) is also a difference of squares: \[ x^2 - 9 = (x - 3)(x + 3) \]

Putting it all together, we get: \[ (x^2 - 4)(x^2 - 9) = (x - 2)(x + 2)(x - 3)(x + 3) \]

Question 2:

A) Given that \(x - 1\) is a factor of the polynomial \(x^3 - 2x^2 - 5x + 6\), we can use synthetic division to divide the polynomial by \(x - 1\).

1. Setting up synthetic division with \(1\):
   • Coefficients: \(1, -2, -5, 6\)

\[ \begin{array}{r|rrrr} 1 & 1 & -2 & -5 & 6 \ & & 1 & -1 & -6 \ \hline & 1 & -1 & -6 & 0 \ \end{array} \]

The quotient from synthetic division is \(x^2 - x - 6\).

2. Next, we can factor \(x^2 - x - 6\): \[ x^2 - x - 6 = (x - 3)(x + 2) \]

So the complete factorization of \(x^3 - 2x^2 - 5x + 6\) is: \[ (x - 1)(x - 3)(x + 2) \]

Question 3:

A) The function \(f(x) = 3(2x - 1)(x + 2)(8x - 5)^2\) has x-intercepts at the points where \(f(x) = 0\).

The x-intercepts can be found by setting each factor to zero:

1. \(2x - 1 = 0 \Rightarrow x = \frac{1}{2}\)
2. \(x + 2 = 0 \Rightarrow x = -2\)
3. \((8x - 5)^2 = 0 \Rightarrow 8x - 5 = 0 \Rightarrow x = \frac{5}{8}\) (This factor has a multiplicity of 2.)

Thus, the x-intercepts are:

• \(x = \frac{1}{2}\)
• \(x = -2\)
• \(x = \frac{5}{8}\)

This means there are three distinct x-intercepts.

Question 4:

A) To find the zeros of the polynomial \(j(x) = 64x^4 + 160x^3 + 48x^2 - 40x - 16\), we would typically analyze the graph. However, I cannot view images. Given that it states there are three inflection points, it’s likely that the polynomial has 4 total roots due to its degree.

Using a numeric approach such as the Rational Root Theorem, or factoring (if possible), we can typically find estimates for the roots. If you have the graph or additional data, please share those values or share graphs, and we can identify roots directly.

Question 5:

A) For \(q(x)\) given as \(q(x) = (x + 4)(x + _)(x - _\), we would again require specifics from the graph; however, given a polynomial with 2 turning points generally indicates a cubic function.

If we could derive the location of the turning points and potential zeros of the function from the visual or numerical points on the graph, we can assume some reasonable endpoints.

Please share a description of the graph, including where it appears to intersect the x-axis (yielding the zeros) for accurate completion of this polynomial factorization request.