High School

13. Factorize the following functions, find the zeroes, and then sketch their graphs.

a. [tex]f(x) = x^4 - 4x^3 - 7x^2 + 22x + 24[/tex]

b. [tex]f(x) = x^4 - x^3 - 35x^2 + 57x + 90[/tex]

c. [tex]f(x) = x^5 - 2x^4 - 13x^3 + 38x^2 - 24x[/tex]

Answer :

Let's tackle these three functions one by one, factorizing them, finding the zeroes, and then I'll guide how to sketch the graphs.

a. Function: [tex]f(x) = x^4 - 4x^3 - 7x^2 + 22x + 24[/tex]

  1. Factorization:
    To factor this polynomial, we can use techniques like synthetic division, trial and error with potential rational roots, or recognizing patterns.

    First, try to find rational roots by testing possible values using the Rational Root Theorem, which suggests that potential roots include factors of the constant term (24) over the leading coefficient (1). Hence, possible roots are [tex]\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24[/tex].

    Upon testing, [tex]x = 1[/tex] is a root. Use synthetic division to divide the polynomial by [tex]x - 1[/tex]:

    [tex](x^4 - 4x^3 - 7x^2 + 22x + 24) \div (x - 1)[/tex]
    yields [tex]x^3 - 3x^2 - 10x + 24[/tex].

    Repeat the process to factor [tex]x^3 - 3x^2 - 10x + 24[/tex]. You’ll find another factor [tex]x = 2[/tex].

    Continuing the process, you find the full factorization:
    [tex]f(x) = (x - 1)(x - 2)(x + 3)(x - 4)[/tex]

  2. Zeroes:
    Set each factor to zero:

    • [tex]x - 1 = 0 \Rightarrow x = 1[/tex]
    • [tex]x - 2 = 0 \Rightarrow x = 2[/tex]
    • [tex]x + 3 = 0 \Rightarrow x = -3[/tex]
    • [tex]x - 4 = 0 \Rightarrow x = 4[/tex]

  3. Sketch:

    • Plot these roots [tex]1, 2, -3, 4[/tex] on the x-axis.
    • The function is a quartic (degree 4), so its ends point up.
    • The critical values divide the x-axis into intervals. Test points in each interval to see if the function is positive or negative.

b. Function: [tex]f(x) = x^4 - x^3 - 35x^2 + 57x + 90[/tex]

  1. Factorization and Zeroes:
    Follow a similar process by testing rational roots.
    Testing reveals that [tex]x = 2[/tex] is a root:
    [tex](x^4 - x^3 - 35x^2 + 57x + 90) \div (x - 2)[/tex]
    gives [tex]x^3 + x^2 - 33x - 45[/tex].

    Further testing shows [tex]x = -3[/tex] is another root:
    [tex]x^3 + x^2 - 33x - 45 \div (x + 3)[/tex] gives [tex]x^2 - 2x - 15[/tex].

    This factors as [tex](x - 5)(x + 3)[/tex].
    Thus, [tex]f(x) = (x - 2)(x + 3)^2(x - 5)[/tex].

  2. Zeroes:

    • [tex]x = 2[/tex], [tex]x = -3[/tex] (repeated), [tex]x = 5[/tex]

  3. Sketch:

    • Major zeroes are [tex]2, -3, 5[/tex].
    • Double root at [tex]x = -3[/tex] indicates the graph touches and turns around there, creating local minima.

c. Function: [tex]f(x) = x^5 - 2x^4 - 13x^3 + 38x^2 - 24x[/tex]

  1. Factorization and Zeroes:
    First, factor out [tex]x[/tex] as it’s common:
    [tex]x(x^4 - 2x^3 - 13x^2 + 38x - 24)[/tex]

    Further factorization via testing or synthetic division reveals:

    • [tex]x = 3[/tex], [tex]x = 4[/tex], [tex]x = 2[/tex], leading to
    • [tex]f(x) = x(x - 3)(x - 1)^2(x - 2)[/tex]

  2. Zeroes:

    • [tex]x = 0[/tex], [tex]x = 3[/tex], [tex]x = 1[/tex] (repeated), [tex]x = 2[/tex]

  3. Sketch:

    • The graph passes through the origin ([tex]x = 0[/tex]).
    • Ends of the graph point in opposite directions due to odd degree.

These steps provide a basic framework to understand the factorization, zeroes, and general shape of the graphs for each function. For accurate graph plotting, use graphing tools or calculators to visualize specific details.

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