Polynomials 3 (1 Viewer)

coyazayo · Apr 17, 2016

:0 Hii everyone. Just a polynomial question id like to ask. Thanx in advance

P(x) is an odd polynomial of degree 3. It has x+4 as a factor, and when it is divided by x-3 the remainder is 21. Find P(x)

InteGrand · Apr 17, 2016

coyazayo said:
:0 Hii everyone. Just a polynomial question id like to ask. Thanx in advance

P(x) is an odd polynomial of degree 3. It has x+4 as a factor, and when it is divided by x-3 the remainder is 21. Find P(x)

$\noindent Here is one way to do it.$

$\noindent Since $P$ is an odd polynomial of degree 3, it only contains odd-power terms (you may like to prove this as an exercise, it is true for general degreed polynomials too, apart from the trivial case of the zero polynomial). So $P(x) = ax^3 + bx$ for some real $a,b$.$

$\noindent By the factor theorem, $P(-4) = 0$, so $a\left(-4\right)^3 -4b = 0 \Longleftrightarrow -64a -4b = 0 \Longleftrightarrow 16 a + b = 0$. Also, $P(3)=21$ by the remainder theorem, so $a\times 3^3 + 3b = 21 \Longleftrightarrow 27a + 3b =21 \Longleftrightarrow 9a + b = 7$. We now have two equations in two unknowns ($a$ and $b$) that we can solve.$

Polynomials 3 (1 Viewer)

coyazayo

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InteGrand

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