Elementary Algebra

Concept	Quick Reference
Square of a sum	$(A+B{)}^2=A^2+2AB+B^2$
Difference of squares	$A^2-B^2=(A-B)(A+B)$
Completing the square	$a{x}^2+bx+c=a(x+h{)}^2+k$, where $h=\frac{b}{2a}$
Quadratic formula	$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Important Identities

The following identities are used constantly in calculus for expanding, factoring, and simplifying expressions. You should know them fluently.

1. $(A+B{)}^2=A^2+2AB+B^2$     (Square of a Sum)
2. $(A-B{)}^2=A^2-2AB+B^2$     (Square of a Difference)
3. $(A+B{)}^3=A^3+3A^{2}B+3A{B}^2+B^3$     (Cube of a Sum)
4. $(A-B{)}^3=A^3-3A^{2}B+3A{B}^2-B^3$     (Cube of a Difference)
5. $A^2-B^2=(A-B)(A+B)$     (Difference of Squares)
6. $A^3-B^3=(A-B)(A^2+AB+B^2)$     (Difference of Cubes)
7. $A^3+B^3=(A+B)(A^2-AB+B^2)$     (Sum of Cubes)
8. $A^n-B^n=(A-B)(A^{n-1}+A^{n-2}B+\cdots +A{B}^{n-2}+B^{n-1})$     (Difference of $n$th Powers)
9. $A^n+B^n=(A+B)(A^{n-1}-A^{n-2}B+A^{n-3}{B}^2-\cdots -A{B}^{n-2}+B^{n-1})$    ($n$ odd)     (Sum of $n$th Powers)

The Binomial Theorem

Identities (1) through (4) above are special cases of the binomial theorem. If $n$ is a positive integer, then

$$(A+B{)}^n=A^n+n{A}^{n-1}B+\frac{n(n-1)}{2!}{A}^{n-2}{B}^2+\frac{n(n-1)(n-2)}{3!}{A}^{n-3}{B}^3+\cdots +nA{B}^{n-1}+B^n$$

where $m!=m\cdot (m-1)\cdot (m-2)\cdots 3\cdot 2\cdot 1$. This can also be written using binomial coefficients:

$$(A+B{)}^n=(\genfrac{}{}{0ex}{}{n}{0})A^n{B}^0+(\genfrac{}{}{0ex}{}{n}{1})A^{n-1}{B}^1+\cdots +(\genfrac{}{}{0ex}{}{n}{k})A^{n-k}{B}^k+\cdots +(\genfrac{}{}{0ex}{}{n}{n-1})A^1{B}^{n-1}+(\genfrac{}{}{0ex}{}{n}{n})A^0{B}^n$$

where

$$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!}=\frac{n(n-1)\cdots (n-k+1)}{k(k-1)\cdots 3\cdot 2\cdot 1}.$$

By convention, $0! = 1$.

Completing the Square

We can rewrite any quadratic expression $a{x}^2+bx+c$ in the form $a(x+h{)}^2+k$, where

$$h=\frac{b}{2a},\text{and}k=c-a{h}^2=c-\frac{b^2}{4a}.$$

This technique is called completing the square. It is used in calculus to evaluate certain integrals and to find the vertex form of a parabola.

Quadratic Equations

In the quadratic equation $a{x}^2+bx+c=0$ (with $a\ne 0$), the solutions (also called roots) are given by the quadratic formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

The expression $b^2-4ac$ is called the discriminant. Its sign determines the nature of the roots:

• If , the roots are distinct and real.
• If $b^2-4ac=0$, the roots are real and equal (a repeated root).
• If , the roots are complex (non-real).

When a quadratic is reduced to the form $x^2+px+q=0$ and has two roots $x=r_1$ and $x=r_2$, we can factor it as

$$x^2+px+q=(x-r_1)(x-r_2).$$

Expanding the right-hand side reveals two useful relations between the coefficients and the roots:

• $p=-(r_1+r_2)$     (sum of roots, with sign changed)
• $q=r_1{r}_2$     (product of roots)

Frequently Asked Questions

Why do I need the binomial theorem in calculus?

The binomial theorem appears when computing derivatives from the definition. For example, computing the derivative of $f(x)=x^n$ using the limit definition requires expanding $(x+h{)}^n$, and the binomial theorem is the tool for doing that systematically.

When should I complete the square instead of using the quadratic formula?

Completing the square is preferred when you need the vertex form of a parabola, when evaluating integrals of the form $\int \frac{dx}{a{x}^2+bx+c}$, or when you want to understand the geometric meaning of the roots. The quadratic formula is faster when you just need the numerical values of the roots.

What does the difference of cubes formula have to do with calculus?

The difference of cubes and more generally the difference of $n$th powers formula is used when computing derivatives of power functions from the definition. The factorization $A^n-B^n=(A-B)(A^{n-1}+\cdots +B^{n-1})$ allows you to cancel the factor $A-B$ that appears in the denominator of the difference quotient $\frac{f(x+h)-f(x)}{h}$.