4) create an equation with the following roots: -4 , 4 , -3 , 2/5 (fraction) 5) create an equation with the following roots: ±

Question

4) create an equation with the following roots: -4 , 4 , -3 , 2/5 (fraction)

5) create an equation with the following roots: ± 2i , -7 , 1

6) find all the zeros on the following polynomial: 2x^3 – 11x^2 – 16x + 105 Given zero: 5

in progress 0
Harper 1 week 2021-10-04T05:17:20+00:00 1 Answer 0

Answers ( )

    0
    2021-10-04T05:18:42+00:00

    Answer:

    4. x^4+\dfrac{13}{5}x^3-\dfrac{86}{5}x^2-\dfrac{208}{5}x+\dfrac{96}{5}

    5. x^4+6x^3-3x^2+24x-28

    6. 5, -3, \dfrac{7}{2}

    Step-by-step explanation:

    4. If numbers -4 , 4 , -3 , \dfrac{2}{5} are roots of the equation, then

    x-(-4)=x+4\\ \\x-4\\ \\x-(-3)=x+3\\ \\x-\dfrac{2}{5}

    are factors of the equation. The equation, therefore, can be written in the following way:

    (x+4)(x-4)(x+3)\left(x-\dfrac{2}{5}\right)\\ \\=(x^2-16)\left(x^2-\dfrac{2}{5}x+3x-\dfrac{6}{5}\right)\\ \\=(x^2-16)\left(x^2+\dfrac{13}{5}x-\dfrac{6}{5}\right)\\ \\=x^4+\dfrac{13}{5}x^3-\dfrac{6}{5}x^2-16x^2-\dfrac{208}{5}x+\dfrac{96}{5}\\ \\=x^4+\dfrac{13}{5}x^3-\dfrac{86}{5}x^2-\dfrac{208}{5}x+\dfrac{96}{5}

    5. If numbers \pm 2i, -7, 1 are roots of the equation, then

    x-2i\\ \\x+2i\\ \\x-(-7)=x+7\\ \\x-1

    are factors of the equation. The equation, therefore, can be written in the following way:

    (x-2i)(x+2i)(x+7)(x-1)\\ \\=(x^2+4)(x^2-x+7x-7)\\ \\=(x^2+4)(x^2+6x-7)\\ \\=x^4+6x^3-7x^2+4x^2+24x-28\\ \\=x^4+6x^3-3x^2+24x-28

    6. Given polynomial

    2x^3 - 11x^2 - 16x + 105

    and its zero 5, then

    2x^3 - 11x^2 - 16x + 105\\ \\=2x^3 -10x^2 -x^2 +5x-21x+105\\ \\=2x^2 (x-5)-x(x-5)-21(x-5)\\ \\=(x-5)(2x^2 -x-21)\\ \\=(x-5)(2x^2 +6x-7x-21)\\ \\=(x-5)(2x(x+3)-7(x+3))\\ \\=(x-5)(x+3)(2x-7)

    Therefore, zeros are

    5, -3, \dfrac{7}{2}

Leave an answer

45:7+7-4:2-5:5*4+35:2 =? ( )