Solve a polynomial given the value of another polynomial

Question

If $$x^2−9x+5=0$$ then what is $$x^4−18x^3+81x^2+42$$

I know I can simply find what x is and plug it in, but given this is a practice question on a problem solving paper I know there is a 'smart' way to do it.

Can someone give a hint to how to approach these questions?

Answer 1

\begin{align} x^2−9x+5=0 &\implies x^2 = 9x - 5 \\ &\implies x^3 = 9x^2 - 5x = 76x - 45\\ &\implies x^4 = 76x^2-45x = 639x - 380 \\ \hline x^4−18x^3+81x^2+42 &= \cdots \end{align}

OR

\begin{array}{r} &&&&&& x^2 &-9x &-5 \\ &&&&--- &--- &--- &--- &--- \\ x^2 &-9x &+5 &\mid &x^4 &-18x^3 &+81x^2 & +0x &+42 \\ &&&&x^4 & -9x^3 & +5x^2 \\ &&&&--- &--- &--- \\ &&&&& -9x^3 &+76x^2 & +0x \\ &&&&& -9x^3 &+81x^2 &-45x \\ &&&&&--- &--- &--- \\ &&&&&& -5x^2 &+45x &+42 \\ &&&&&& -5x^2 &+45x &-25 \\ &&&&&&--- &--- &--- \\ &&&&&&&& 67 \\ \end{array}

So $ x^4−18x^3+81x^2+42 = (x^2−9x-5)(x^2−9x+5) + 67 = \cdots$

Answer 2

Hint: the smart way is to find a polynomial of $x^2 - 9x + 5$ which results in $x^4 - 18x^3 + 81x^2 + 42$. Try: $$ (x^2 - 9x + 5)^2 + \text{_______} \cdot (x^2 - 9x + 5) + \text{_______}. $$ where you have to fill in the blanks.

Answer 3

What I would do is use the first equation to get an expression first of $x^4$ in terms of the lower terms, then of $x^3$ similarily. Substituing in the second equation successively reduces the degree of the expression you want to compute, and may end up simplifying the whole thing. For the detailed process, see my spoiler below. This method may not be the most efficient, but it worked out without much efforts.

Multiply the first equation by $x^2$: you will get an expression for $x^4$, that you can then substitute in the second. That is, we have $$x^4=9x^3-5x^2$$ according to the first equation, so that $$x^4-18x^3+81x^2+42=-9x^3+76x^2+42$$ Now, multiply the first equation by $x$ to get an expression of $x^3$: $$x^3=9x^2-5x$$ and hence, substituing in the second equation, we obtain $$-9x^3+76x^2+42=-9(9x^2-5x)+76x^2+42=-5x^2+45x+42$$ Now, notice that $-5x^2+45x=5(-x^2+9x)$ which, according to the original first equation, simplifies to $5\times 5 =25$. Hence, we obtain as a final result $$x^4-18x^3+81x^2+42=25+42=67$$

Answer 4

$x^2-9x+5=0$

$\Rightarrow x^2-9x=-5$

$\Rightarrow (x^2-9x)^2 = 25$

$\Rightarrow x^4 - 18x^3 + 81x^2 = 25$

$\Rightarrow x^4 - 18x^3 + 81x^2 + 42 = 67$