if 10 is the remainder obtained on dividing the polynomial p(x) = ax^2 - 3x^2 - 8x +7 by q(x) = 2x+1. find the value of a
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if 10 is the remainder obtained on dividing the polynomial p(x) = ax^2 - 3x^2 - 8x +7 by q(x) = 2x+1. find the value
Step-by-step explanation:
Given:
[tex]p(x) = ax^{2} - 3x^{2} - 8x + 7[/tex]
When divided by 2x + 1 it gives remainder as 10.
So,
2x + 1 = 0
x = -1/2
By substituting,
[tex]a(\frac{-1}{2})^{2} - 3(\frac{-1}{2})^{2} - 8(\frac{-1}{2}) + 7 = 10\\\\\frac{a}{4} - \frac{3}{4} + 4 + 7 = 10\\ \\\frac{a - 3}{4} = -1\\\\a - 3 = -4 \\\\a = -4 + 3 \\\\a = -1[/tex]
[tex]2x + 1 = 0 \\ \\ 2x = - 1 \\ \\ x = \frac{ - 1}{2} [/tex]
[tex] = a {x}^{2} - 3 {x}^{2} - 8x + 7 \\ \\ = a( \frac{ - 1}{2} ) {}^{2} - 3( \frac{1}{2} ) {}^{2} - 8( \frac{ - 1}{2} ) + 7 \\ \\ = \frac{a}{2} - \frac{3}{4} + \frac{8}{2} + \frac{7}{1} \\ \\ = \frac{2a - 3 + 16 + 28}{4} \\ \\ = \frac{2a + 41}{4} \\ \\ = 2( \frac{a + 41}{2} ) \\ \\ a + 41 = 2 \\ \\ a = 41 - 2 \\ \\ a = 39[/tex]