Urgent! The polynomials $a x^{3} - 3 x^{2} + 2 x - 3$ $ax^3-3x^2+2x-3$ and $a x^{2} - 5 x + a$ $ax^2-5x+a$ when divided by $x - 2$ $x-2$ leave remainders of $p$ $p$ and $q$ $q$ respectively. Find the value of $a$ $a$ if $p = 3 q$ $p=3q$ . How? Urgent thanks!

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Answer 1 · 2016-11-17T00:47:14

$a = \frac{19}{7}, p = \frac{75}{7}, q = \frac{25}{7}$ $a=19/7,p=75/7,q=25/7$

Calling

$f_{1} (x) = a x^{3} - 3 x^{2} + 2 x - 3$ $f_1(x)=ax^3-3x^2+2x-3$
$f_{2} (x) = a x^{2} - 5 x + a$ $f_2(x)=ax^2-5x+a$

we know that

$f_{1} (x) = q_{1} (x) (x - 2) + p$ $f_1(x)=q_1(x)(x-2)+p$ and
$f_{2} (x) = q_{2} (x) (x - 2) + q$ $f_2(x)=q_2(x)(x-2)+q$

so

$f_{1} (2) = 8 a - 12 + 4 - 3 = p$ $f_1(2) = 8a-12+4-3=p$
$f_{2} (2) = 4 a - 10 + a = q$ $f_2(2)=4a-10+a=q$ and also
$p = 3 q$ $p=3q$

Solving

${(8a-11=p),(5a-10=q),(p=3q):}$

we obtain

$a=19/7,p=75/7,q=25/7$

Urgent! The polynomials ax^3-3x^2+2x-3ax3−3x2+2x−3ax^3-3x^2+2x-3 and ax^2-5x+aax2−5x+aax^2-5x+a when divided by x-2x−2x-2 leave remainders of ppp and qqq respectively. Find the value of aaa if p=3qp=3qp=3q. How? Urgent thanks!