Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. match each expression on the left with its quotient on the right. $(x^2 - 3x - 18) div (x + 3)$ $x - 6$ $(x^3 - x^2 - 5x - 3) div (x^2 + 2x + 1)$ $x^2 - x + 1$ $(x^3 - 4x^2 + 4x - 3) div (x - 3)$ $x - 3$ clear click and hold an item in one column, then drag it to the matching item in the other column. be sure y target before releasing. the target will highlight or the cursor will change. need help? watch this video

Explanation:

Step1: Solve \((x^2 - 3x - 18) \div (x + 3)\)

Factor the numerator: \(x^2 - 3x - 18=(x - 6)(x + 3)\). Then divide by \((x + 3)\): \(\frac{(x - 6)(x + 3)}{x + 3}=x - 6\) (assuming \(x
eq - 3\)).

Step2: Solve \((x^3 - x^2 - 5x - 3) \div (x^2 + 2x + 1)\)

Factor the denominator: \(x^2+2x + 1=(x + 1)^2\). Use polynomial long - division or synthetic division. Let's try to factor the numerator. We can use trial and error. If we assume the numerator factors as \((x + 1)(ax^2+bx + c)\), then \((x + 1)(ax^2+bx + c)=ax^3+(b + a)x^2+(c + b)x + c\). Comparing with \(x^3 - x^2 - 5x - 3\), we have \(a = 1\), \(b+a=-1\Rightarrow b=-2\), \(c + b=-5\Rightarrow c=-3\), \(c=-3\). So the numerator is \((x + 1)(x^2-2x - 3)=(x + 1)(x - 3)(x+1)=(x + 1)^2(x - 3)\). Then \(\frac{(x + 1)^2(x - 3)}{(x + 1)^2}=x - 3\) (assuming \(x
eq - 1\)).

Step3: Solve \((x^3 - 4x^2 + 4x - 3) \div (x - 3)\)

Use polynomial long - division. Divide \(x^3-4x^2 + 4x - 3\) by \(x - 3\).
\(x^3\div x=x^2\), multiply \((x - 3)\) by \(x^2\) to get \(x^3-3x^2\). Subtract from the dividend: \((x^3-4x^2 + 4x - 3)-(x^3-3x^2)=-x^2+4x - 3\).
\(-x^2\div x=-x\), multiply \((x - 3)\) by \(-x\) to get \(-x^2+3x\). Subtract: \((-x^2+4x - 3)-(-x^2+3x)=x - 3\).
\(x\div x = 1\), multiply \((x - 3)\) by \(1\) to get \(x - 3\). Subtract: \((x - 3)-(x - 3)=0\). So the quotient is \(x^2-x + 1\).

Answer:

\((x^2 - 3x - 18)\div(x + 3)\) matches with \(x - 6\)

\((x^3 - x^2 - 5x - 3)\div(x^2 + 2x + 1)\) matches with \(x - 3\)

\((x^3 - 4x^2 + 4x - 3)\div(x - 3)\) matches with \(x^2 - x + 1\)