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<span><u><em>The correct answer is: </em></u>
x</span>²<span>-3x-4 -</span>
<span>.
<u><em>Explanation</em></u><span><u><em>: </em></u>
To divide these polynomials, the dividend, x</span></span>³<span><span>-5x</span></span>²<span><span>+2x+5, goes under the box, and the divisor, x-2, goes outside of the box.
We first want to find what we multiply x by to make x</span></span>³<span><span>; x*x</span></span>²<span><span>=x</span></span>³<span><span>, so we write x</span></span>²<span><span> over -5x</span></span>²<span><span> in the box.
<u>Now we multiply our divisor, x-2, by x</u></span></span><u>²</u><span><span><u>: </u>
x*x</span></span>²<span><span>=x</span></span>³<span><span>
-2*x</span></span>²<span><span>=-2x</span></span>²<span><span>.
This goes under x</span></span>³<span><span>-5x</span></span>²<span><span> in the box.
<u>Now we subtract: </u>
(x</span></span>³<span><span>-5x</span></span>²<span><span>)-(x</span></span>³<span><span>-2x</span></span>²<span><span>)=-3x</span></span>²<span><span>.
This goes below the 2x</span></span>²<span><span> we wrote, and we bring down 2x.
Now we see what we multiply x by to make -3x</span></span>²<span><span>;
x*-3x=-3x</span></span>²<span><span>, so we write -3x beside x</span></span>²<span><span> in the quotient (at the top).
<u>We multiply our divisor, x-2, by -3x: </u>
x*-3x=-3x</span></span>²
<span><span>-2*-3x=6x.
We write -3x</span></span>²<span><span>+6x under our -3x</span></span>²<span><span>+2x and <u>subtract</u>:
(-3x</span></span>²<span><span>+2x)-(-3x</span></span>²<span><span>+6x)=-4x.
Write this below 6x. Bring down +5.
Now we see what we multiply x by to make -4x;
x*-4=-4x, so write -4 beside -3x in the quotient.
<u>Multiply our divisor, x-2, by -4: </u>
x*-4=-4x;
-2*-4=8;
write -4x+8 below -4x+5 and <u>subtract:</u>
(-4x+5)-(-4x+8)=-3.
This gives us the quotient x</span></span>²<span><span>-3x-4 with a remainder of -3, so we write our full answer as x</span></span>²<span><span>-3x-4-(3/x-2).</span></span>
x</span>²<span>-3x-4 -</span>
<u><em>Explanation</em></u><span><u><em>: </em></u>
To divide these polynomials, the dividend, x</span></span>³<span><span>-5x</span></span>²<span><span>+2x+5, goes under the box, and the divisor, x-2, goes outside of the box.
We first want to find what we multiply x by to make x</span></span>³<span><span>; x*x</span></span>²<span><span>=x</span></span>³<span><span>, so we write x</span></span>²<span><span> over -5x</span></span>²<span><span> in the box.
<u>Now we multiply our divisor, x-2, by x</u></span></span><u>²</u><span><span><u>: </u>
x*x</span></span>²<span><span>=x</span></span>³<span><span>
-2*x</span></span>²<span><span>=-2x</span></span>²<span><span>.
This goes under x</span></span>³<span><span>-5x</span></span>²<span><span> in the box.
<u>Now we subtract: </u>
(x</span></span>³<span><span>-5x</span></span>²<span><span>)-(x</span></span>³<span><span>-2x</span></span>²<span><span>)=-3x</span></span>²<span><span>.
This goes below the 2x</span></span>²<span><span> we wrote, and we bring down 2x.
Now we see what we multiply x by to make -3x</span></span>²<span><span>;
x*-3x=-3x</span></span>²<span><span>, so we write -3x beside x</span></span>²<span><span> in the quotient (at the top).
<u>We multiply our divisor, x-2, by -3x: </u>
x*-3x=-3x</span></span>²
<span><span>-2*-3x=6x.
We write -3x</span></span>²<span><span>+6x under our -3x</span></span>²<span><span>+2x and <u>subtract</u>:
(-3x</span></span>²<span><span>+2x)-(-3x</span></span>²<span><span>+6x)=-4x.
Write this below 6x. Bring down +5.
Now we see what we multiply x by to make -4x;
x*-4=-4x, so write -4 beside -3x in the quotient.
<u>Multiply our divisor, x-2, by -4: </u>
x*-4=-4x;
-2*-4=8;
write -4x+8 below -4x+5 and <u>subtract:</u>
(-4x+5)-(-4x+8)=-3.
This gives us the quotient x</span></span>²<span><span>-3x-4 with a remainder of -3, so we write our full answer as x</span></span>²<span><span>-3x-4-(3/x-2).</span></span>