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Quiz 3
1. Multiply and simplify: $(x+2)(x^2-x+2)$.
Solution: Compute using the distributive law $a(b+c)=ab+ac$ repeatedly:
$$\begin{array}{ll}
(x+2)(x^2-x+2) &= (x+2)x^2 - (x+2)(x)+(x+2)(2) \\
&= (x^3+2x^2) - (x^2+2x) + (2x+4) \\
&= x^3+ x^2+0x+4 \\
&= x^3+x^2+4.
\end{array}$$
2. Perform the long division $(x^2+7x+12) \div (x+3)$.
Solution: See the following image:
This means that the answer to the calculation is
$$(x^2+7x+12) \div (x+3)=x+4.$$
3. Multiply and simplify: $(2x+1)^2$.
Solution: Compute using the distributive law
$$\begin{array}{ll}
(2x+1)^2 &= (2x+1)(2x+1) \\
&= (2x+1)(2x) + (2x+1)(1) \\
&= (4x^2+2x) + (2x+1) \\
&= 4x^2 +4x + 1.
\end{array}$$
4. Solve the equation $x+5=10$.
Solution: Add $-5$ to both sides to get
$$x+5+(-5) = 10+(-5),$$
and now simplify both sides to get
$$x=5,$$
which is the solution.
5. Solve the equation $2(x+1)=3(x-2)$.
Solution: First distribute the $2$ on the left into the sum and distribute the $3$ on the right to get
$$2x+2 = 3x-6.$$
Now add $-2x$ to both sides to get all the $x$'s on one side:
$$2x+2+(-2x) = 3x-6+(-2x),$$
and simplify to get
$$2 = x-6.$$
Now add $6$ to both sides to get
$$2+6 = x-6+6,$$
and simplify to get
$$8=x,$$
which is the solution.
