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Answer 1 · 2018-05-29T15:31:53

$g (x) \to - 3 f (x) - 6 x^{3} + 6 x^{2} + 15$ $g(x)->-3f(x) -6x^3+6x^2+15$

Explanation:

Given: $f (x) = 2 x^{5} - x^{3} + x^{2} + 4$ $f(x)=2x^5-x^3+x^2+4$

Target to achieve: $g (x) = - 6 x^{5} - 3 x^{3} + 3 x^{2} + 3$ $g(x)=color(green)(-6x^5-3x^3+3x^2+3)$

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Consider the first term in $f (x)$ $f(x)$ which is $2 x^{5}$ $2x^5$

We need to change this into $- 6 x^{5}$ $-6x^5$ and this can be achieved by using: $(- 3) \times f (x)$ $(-3)xxf(x)$ It will change all the other terms in $f (x)$ $f(x)$ but we can adjust them as we go along.

$+ 2 x^{5} - x^{3} + d x^{2} + 4$ $+2x^5-x^3+color(white)("d")x^2+4$
$ul(color(white)("ddddddddddddd.d")-3)larr"Multiply"$
$-6x^5+3x^3-3x^2-12$

We now have the target $x^5$ term $->-3f(x)$ but the rest does not match the target. Thus we determine the adjustment by applying a subtraction.

Target $color(white)(".")->-6x^5-3x^3+3x^2+3$
$-3f(x) ->ul(-6x^5+3x^3-3x^2-12larr" Subtract")$
$color(white)("dddddddddddd")color(red)(0-6x^3+6x^2+15)$

Giving $color(white)("d")-3f(x)color(red)(-6x^3+6x^2+15larr" The adjustment")$

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$color(blue)("Check")$

$-3f(x)->color(white)("dddddd")-6x^5+3x^3-3x^2-12$
The adjustment $-> ul(color(white)("dddddd")color(red)(-6x^3+6x^2+15)larr" Add")$
$color(white)("dddddddddddddddd")-6x^5-3x^3+3x^2+3$
The target $->color(white)("ddddd")color(green)(-6x^5-3x^3+3x^2+3)$

A perfect match so $-3f(x) -6x^3+6x^2+15$ is correct

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