The polynomial 3x² + mx? – nx – 10 has a factor of (x – 1). When divided by x + 2, the remainder is 36. What are the values of m and n

Question

The polynomial 3x² + mx? – nx – 10 has a factor of (x – 1). When divided by x + 2, the remainder is 36. What are
the values of m and n?

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Khoii Minh 5 years 2021-07-26T02:47:55+00:00 1 Answers 21 views 0

Answers ( )

  1. RuslanHeatt
    0
    2021-07-26T02:49:42+00:00
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    Answer:

    m = 12

    n =3

    Step-by-step explanation:

    Given

    P(x) = x^3 + mx^2 - nx - 10

    Required

    The values of m and n

    For x – 1;

    we have:

    x - 1 = 0

    x=1

    So:

    P(1) = (1)^3 + m*(1)^2 - n*(1) - 10

    P(1) = 1 + m*1 - n*1 - 10

    P(1) = 1 + m - n - 10

    Collect like terms

    P(1) = m - n + 1 - 10

    P(1) = m - n -9

    Because x – 1 divides the polynomial, then P(1) = 0;

    So, we have:

    m - n -9 = 0

    Add 9 to both sides

    m - n = 9 — (1)

    For x + 2;

    we have:

    x + 2 = 0

    x = -2

    So:

    P(-2) = (-2)^3 + m*(-2)^2 - n*(-2) - 10

    P(-2) = -8 + 4m + 2n - 10

    Collect like terms

    P(-2) = 4m + 2n - 10 - 8

    P(-2) = 4m + 2n - 18

    x + 2 leaves a remainder of 36, means that P(-2) = 36;

    So, we have:

    4m + 2n - 18 = 36

    Collect like terms

    4m + 2n = 36+18

    4m + 2n = 54

    Divide through by 2

    2m + n=27 — (2)

    Add (1) and (2)

    m + 2m - n + n = 9 +27

    3m =36

    Divide by 3

    m = 12

    Substitute m = 12 in (1)

    m - n =9

    Make n the subject

    n = m - 9

    n = 12 - 9

    n =3

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