01/14 1. Algebraic Foundations and Factoring Free Online FlashCards

How do rational and irrational numbers differ?

Rational numbers can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b\ne0$. Irrational numbers cannot be written as a ratio of integers.

What does absolute value represent?

Absolute value is a number’s distance from zero on the number line, so it is always nonnegative. For example, $|-5|=5$.

State the distributive property.

The distributive property states $a(b+c)=ab+ac$. In reverse, it supports factoring, such as $3x+12=3(x+4)$.

What makes two terms like terms?

Like terms have exactly the same variables raised to the same powers. Only like terms can be combined, such as $5x+2x=7x$.

Evaluate $3x^2-4x+1$ when $x=-2$.

Substitute $-2$ for $x$: $3(-2)^2-4(-2)+1=12+8+1=21$.

Simplify $18-2[3^2-(4+1)]$.

The simplified value is $10$: $18-2[3^2-(4+1)]=18-2(9-5)=18-8=10$.

Why do $-3^2$ and $(-3)^2$ differ?

$-3^2=-9$ because the exponent applies before the leading negative sign. In contrast, $(-3)^2=9$ because the negative is inside the base.

Solve $4x-7=21$.

The solution is $x=7$. Add $7$ to both sides to get $4x=28$, then divide by $4$.

What does factoring do?

Factoring rewrites an expression as a product of factors; it reverses multiplication. For example, $ab+ac=a(b+c)$.

Factor $12x^3+18x^2$.

The GCF is $6x^2$, so $12x^3+18x^2=6x^2(2x+3)$.

Factor $x^3+3x^2+2x+6$ by grouping.

Grouping gives $x^3+3x^2+2x+6=x^2(x+3)+2(x+3)=(x+3)(x^2+2)$.

Factor $x^2+7x+12$.

The factors are $(x+3)(x+4)$, because $3\cdot4=12$ and $3+4=7$.