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The polynomial f(x) is defined by

$$f(x) = 12x^3+25x^2 -4x -12$$

(i) Show that f(-2) = 0 and factorise f(x) completely.

Which i did and got $(x+2)(3x-2)(4x+3)$

(ii) Given that $$12 * 27^y + 25 * 9^y -4 * 3^y -12=0$$

state the value of 3^y and hence find y correct to 3 s.f.

I'm a little stuck on part (ii), what exactly is x being replaced by? I initially thought it was $3^y$ but then $x^3$ would by 3^(3y) instead of $27^y$ , wouldn't it?

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2 Answers 2

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$$27^y = (3^3)^y = 3^{3 \cdot y} = 3^{y \cdot 3} = (3^y)^3$$

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Hint Note that $27^y=(3^y)^3$, $9^y= (3^y)^2$, if $x=3^y$ then:

$$12 \cdot 27^y + 25 \cdot 9^y -4 \cdot 3^y -12=0\equiv 12x^3+25x^2 -4x -12=0$$

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