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Math  /  Algebra

QuestionFactor the following expression completely: z3−9z2+8z−72=z^{3}-9 z^{2}+8 z-72= Submit Question

Studdy Solution

STEP 1

1. The expression z3−9z2+8z−72z^3 - 9z^2 + 8z - 72 is a polynomial of degree 3.
2. The factorization involves finding the roots of the polynomial.
3. The polynomial can be factored using methods such as the Rational Root Theorem, synthetic division, and polynomial division.

STEP 2

1. Use the Rational Root Theorem to identify potential rational roots.
2. Test the potential roots to find an actual root.
3. Use synthetic division or polynomial division to divide the polynomial by (z−root)(z - \text{root}).
4. Factor the resulting quadratic polynomial further, if possible.

STEP 3

Use the Rational Root Theorem to list all possible rational roots of z3−9z2+8z−72z^3 - 9z^2 + 8z - 72.
The Rational Root Theorem states that any rational root, p/qp/q, is a factor of the constant term (-72) divided by a factor of the leading coefficient (1).
Possible rational roots: ±1,±2,±3,±4,±6,±8,±9,±12,±18,±24,±36,±72 \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 9, \pm 12, \pm 18, \pm 24, \pm 36, \pm 72

STEP 4

Test the possible rational roots by substituting them into the polynomial z3−9z2+8z−72z^3 - 9z^2 + 8z - 72 to find an actual root.
Testing z=3z=3: 33−9(3)2+8(3)−72=27−81+24−72=−102≠0 3^3 - 9(3)^2 + 8(3) - 72 = 27 - 81 + 24 - 72 = -102 \neq 0 Testing z=2z=2: 23−9(2)2+8(2)−72=8−36+16−72=−84≠0 2^3 - 9(2)^2 + 8(2) - 72 = 8 - 36 + 16 - 72 = -84 \neq 0 Testing z=4z=4: 43−9(4)2+8(4)−72=64−144+32−72=−120≠0 4^3 - 9(4)^2 + 8(4) - 72 = 64 - 144 + 32 - 72 = -120 \neq 0 Testing z=6z=6: 63−9(6)2+8(6)−72=216−324+48−72=−132≠0 6^3 - 9(6)^2 + 8(6) - 72 = 216 - 324 + 48 - 72 = -132 \neq 0 Testing z=8z=8: 83−9(8)2+8(8)−72=512−576+64−72=−72≠0 8^3 - 9(8)^2 + 8(8) - 72 = 512 - 576 + 64 - 72 = -72 \neq 0 Testing z=1z=1: 13−9(1)2+8(1)−72=1−9+8−72=−72≠0 1^3 - 9(1)^2 + 8(1) - 72 = 1 - 9 + 8 - 72 = -72 \neq 0 Testing z=9z=9: 93−9(9)2+8(9)−72=729−729+72−72=0 9^3 - 9(9)^2 + 8(9) - 72 = 729 - 729 + 72 - 72 = 0
We find that z=9z = 9 is indeed a root.

STEP 5

Use synthetic division to divide the polynomial z3−9z2+8z−72z^3 - 9z^2 + 8z - 72 by (z−9)(z - 9).
Set up the synthetic division:
91−98−7290721080\begin{array}{r|rrrr} 9 & 1 & -9 & 8 & -72 \\ & & 9 & 0 & 72 \\ \hline & 1 & 0 & 8 & 0 \end{array}
The result is z2+8z^2 + 8.

STEP 6

Factor the resulting quadratic polynomial z2+8z^2 + 8.
The quadratic polynomial z2+8z^2 + 8 can be factored over the complex numbers as: z2+8=(z+2i)(z−2i) z^2 + 8 = (z + 2i)(z - 2i)

STEP 7

Combine all the factors to write the complete factorization of the original polynomial.
The complete factorization of z3−9z2+8z−72z^3 - 9z^2 + 8z - 72 is: z3−9z2+8z−72=(z−9)(z+2i)(z−2i) z^3 - 9z^2 + 8z - 72 = (z - 9)(z + 2i)(z - 2i)
Solution: The completely factored form of z3−9z2+8z−72z^3 - 9z^2 + 8z - 72 is: (z−9)(z+2i)(z−2i) (z - 9)(z + 2i)(z - 2i)

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