Question

using remainder theorem, factorize the following equation no 4


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Answer 1

$\bf R eminder \ Theoram \colon \\ \sf If \: a \: polynomial \: p(x) \: is \: divided \\ \sf by \: the \: binomial \: x - 2, \: the \\ \sf reminder \: can \: be \: determined \: by \\ \sf finding \: p(2). \\ \\ \\ \mathfrak{ \huge \underline{ \underline{Solution \colon}}} \\ \\ \sf So, \: looking \: at \: the \: question, \: if \\ \sf p(x) = 4x^3 + 7x^2 - 36x - 63 \\ \sf was \: divided \: by \: x - 2, the \\ \sf remainder \: can \: be \: determined \: by \\ \sf finding \: p(2).$

$\sf p(x) = 4x { }^{3} + 7x {}^{2} - 36x - 63 \\ \sf p(2) = 4 {(2)}^{3} + 7 {(2)}^{2} - 36x - 63 \\ \sf \: \: \: \: \: \: \: \: \: = 4 \times 8 + 7 \times 4 - 36 \times 2 - 63 \\ \sf \: \: \: \: \: \: \: \: \: = 32 + 28 - 72 - 63 \\ \sf \: \: \: \: \: \: \: \: \: = - 75$

Answer 2

Step-by-step explanation:

Given, f(x) = 4x³ + 7x² - 36x - 63

Factors of constant term are: ±1, ±3, ±7, ±9, ±21.

Putting x = -3, we have

f(-3) = 4(-3)³ + 7(-3)² - 36(-3) - 63

       = -108 + 63 + 108 - 63

       = 0.

Hence, (x + 3) is a factor of 4x³ + 7x² - 36x - 63.

Long Division Method:

x + 3) 4x³ + 7x² - 36x - 63 (4x² - 5x - 21

         4x³ + 12x²

         -----------------------------

                  -5x² - 36x - 63

                 -5x² - 15x

          ---------------------------

                            -21x - 63

                           -21x - 63

           -----------------------------

                                     0

∴ 4x³ + 7x² - 36x - 63 = (x + 3)(4x² - 5x - 21)

                                   = (x + 3)(4x² + 7x - 12x - 21)

                                   = (x + 3)[x(4x + 7) - 3(4x + 7)]

                                   = (x + 3)(x - 3)(4x + 7).

Hope it helps!