<p>It is given that&nbsp;$\mathrm{f}\left( x \right)={{x}^{4}}+a{{x}^{3}}+b{{x}^{2}}-2x+1$ and $\mathrm{g}\left( x \right)=3{{x}^{4}}-48$, where&nbsp;$a$ and $b$ are integers.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">If&nbsp;$\mathrm{f}\left( x \right)$ has a remainder of $2x-3$ when it is divided by ${{x}^{2}}-4$, find the value of $a$ and of $b$.</div></li><li><div class="list-item-paragraph">Factorise $\mathrm{g}\left( x \right)$ completely. Hence state the real values of $x$ if&nbsp;$\mathrm{g}\left( x \right)=0$.</div></li><li><div class="list-item-paragraph">Given further that&nbsp;$2\mathrm{f}\left( x \right)-3\mathrm{g}\left( x \right)=\mathrm{Q}\left( x \right)\left( x+2 \right)+\mathrm{R}\left( x \right)$, where&nbsp;$\mathrm{Q}\left( x \right)$ and $\mathrm{R}\left( x \right)$ are the quotient and remainder 	respectively, find $\mathrm{R}\left( x \right)$.</div></li></ol>

It is given that $\mathrm{f}\left( x \right)={{x}^{4}}+a{{x}^{3}}+b{{x}^{2}}-2x+1$ and $\mathrm{g}\left( x \right)=3{{x}^{4}}-48$, where $a$ and $b$ are integers.

If $\mathrm{f}\left( x \right)$ has a remainder of $2x-3$ when it is divided by ${{x}^{2}}-4$, find the value of $a$ and of $b$.

Factorise $\mathrm{g}\left( x \right)$ completely. Hence state the real values of $x$ if $\mathrm{g}\left( x \right)=0$.

Given further that $2\mathrm{f}\left( x \right)-3\mathrm{g}\left( x \right)=\mathrm{Q}\left( x \right)\left( x+2 \right)+\mathrm{R}\left( x \right)$, where $\mathrm{Q}\left( x \right)$ and $\mathrm{R}\left( x \right)$ are the quotient and remainder 	respectively, find $\mathrm{R}\left( x \right)$.

Polynomials, Cubic Equations and Partial Fractions - Higher Order Thinking Questions

Question 98cb876b

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