If (x-2) and (x+2) are factors of 6x^3+ax^2+bx+16, determine a and b, and any remaining factors.?

So clearly this function has zeros at x=-2 and x=2, at least.  

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  • 4 weeks ago
    Favourite answer

    Let f(x) = 6x³ + ax² + bx + 16

    (x - 2) is a factor of f(x). Then, f(2) = 0

    6(2)³ + a(2)² + b(2) + 16 = 0

    2a + b = -32 …… [1]

    (x + 2) is a factor of f(x). Then, f(-2) = 0

    6(-2)³ + a(-2)² + b(-2) + 16 = 0

    2a - b = 16 …… [2]

    [1] + [2]:

    4a = -16

    a = -4

    [1] - [2]:

    2b = -48

    b = -24

    f(x) = 6x³ - 4x² - 24x + 16

    = 6x³ + 12x² - 16x² - 32x + 8x + 16

    = (6x³ + 12x²) - (16x² + 32x) + (8x + 16)

    = 6x²(x + 2) - 16x(x + 2) + 8(x + 2)

    = (x + 2)(6x² - 16x + 8)

    = 2(x + 2)(3x² - 8x + 4)

    = 2(x + 2)(x - 2)(3x - 2)

    The answers:

    a = -4 and b = -24

    Remaining factors: 2 and (3x - 2)

  • Vaman
    Lv 7
    4 weeks ago

    6x^3+ax^2+bx+16=0. x=2 is aroot, Put it 6*8+4a+2b+16=0

    4a+2b=-48-16=-64.

    Now put x=-2

    -48+4a-2b+16=0. This gives 4a-2b=32

    2a+b=-32, 2a-b=16, Add the two. 4a=-16, a=-4,

    For b use the first equation. -8+b=-32. b=-24. Please check the result.

  • Philip
    Lv 6
    4 weeks ago

    Put f(x) = 6x^3+ax^2+bx+16;

    Given : (x-2) & (x+2) are both factors of f(x);

    Therefore (x-2)*(x+2) = (x^2-4) is a factor of f(x);

    Then there exists a linear factor g(x) = (px + q) such that (x^2-4)(px +q) = f(x). Now

    (x^2-4)(px+q) = px^3+qx^2-4px-4q = 6x^3+ax^2+bx+16. Equating coefficients of like

    powers of x gives p = 6, q = a, -4p = b, -4q = 16. Then q = a = -4 and b = -24. Then

    g(x) = (6x-4), f(x) = 6x^3 -4x^2 -24x +16 = (6x-4)(x^2-4) = 2(x+2)(3x-2)(x-2). I have

    deliberately lined up the factors containing the zeros of f(x) in ascending order.  which are -2, 2/3, 2. 

  • (x - 2) * (x + 2) = x^2 - 4

    (x^2 - 4) * (mx + n) = 6x^3 + ax^2 + bx + 16

    mx^3 + nx^2 - 4mx - 4n = 6x^3 + ax^2 + bx + 16

    mx^3 = 6x^3

    m = 6

    nx^2 = ax^2

    n = a

    -4mx = bx

    -4m = b

    -4 * 6 = b

    -24 = b

    -4n = 16

    n = -4

    a = -4

    a = -4 , b = -24

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