Algebra: expand, factor and simplify

8.1Automatic simplification#

Every answer is simplified for you: like terms are combined, powers are merged and common factors cancel.

Example 8.1

Combine like terms

1. Type 2x + 3x.

   2x+3xenter

Example 8.2

Cancel a common factor

1. Type (x²−4) / (x²+4x+4).

   (x^2▶−4)÷(x^2▶+4x+4)enter

Example 8.3

Add powers

1. Type x² × x³.

   x^2▶×x^3▶enter

Example 8.4

Pythagorean identity

$$\sin^{2}x+\cos^{2}x=1$$

1. Type sin(x)² + cos(x)².

   sin(x)^2▶+cos(x)^2▶enter

Example 8.5

Add fractions with letters

1. Type x/2 + x/3 using the fraction template.

   /x▼2▶+/x▼3▶enter

8.2Expand brackets#

expand( multiplies brackets out. You find it in the Algebra menu (3, 3).

Example 8.6

Expand a cube

1. Type expand((x+1)^3).

   expand((x+1)^3▶)enter

Example 8.7

Expand a product

1. Type expand((2x−3)(x+4)).

   expand((2x−3)(x+4))enter

Example 8.8

Expand with two letters

1. Type expand((x+y)²).

   expand((x+y)^2▶)enter

Example 8.9

A difference of squares of brackets

1. Type expand((x+1)²−(x−1)²).

   expand((x+1)^2▶−(x−1)^2▶)enter

8.3Factor polynomials and numbers#

factor( breaks a polynomial into factors over the rationals, and a whole number into primes. The result is exact; if nothing factors, you get the same expression back. cFactor( also uses complex numbers.

Example 8.10

Factor a quadratic

$$x^{2}-5x+6=(x-2)(x-3)$$

1. Type factor(x²−5x+6).

   factor(x^2▶−5x+6)enter

Example 8.11

Factor a fourth power

1. Type factor(x⁴−1).

   factor(x^4▶−1)enter

Example 8.12

Sum of cubes

1. Type factor(x³+1).

   factor(x^3▶+1)enter

Example 8.13

A leading coefficient

1. Type factor(6x²+5x+1).

   factor(6x^2▶+5x+1)enter

Example 8.14

Difference of squares

1. Type factor(x²−y²).

   factor(x^2▶−y^2▶)enter

Example 8.15

Factor with complex numbers

1. Type cFactor(x²+1).

   cfactor(x^2▶+1)enter

8.4Common denominators and parts of a fraction#

Example 8.16

Common denominator

$$\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}$$

1. Type comDenom(1/x + 1/y).

   comdenom(1÷x+1÷y)enter

Example 8.17

Numerator of a fraction

1. Type getNum(3/4).

   getnum(3÷4)enter

Example 8.18

Denominator of a fraction

1. Type getDenom(3/4).

   getdenom(3÷4)enter

Example 8.19

Left side of an equation

1. Type left(x=3).

   left(x=3)enter

Example 8.20

Right side of an equation

1. Type right(x=3).

   right(x=3)enter

8.5Polynomial division and coefficients#

These functions take a polynomial and the variable: polyQuotient(p, q, x), polyRemainder(p, q, x), polyGcd(p, q, x), polyDegree(p, x), polyCoeffs(p, x).

Example 8.21

Quotient

1. Type polyQuotient(x³−1, x−1, x).

   polyquotient(x^3▶−1,x−1,x)enter

Example 8.22

Remainder

1. Type polyRemainder(x³+2, x−1, x).

   polyremainder(x^3▶+2,x−1,x)enter

Example 8.23

Greatest common divisor

1. Type polyGcd(x²−1, x²−x, x).

   polygcd(x^2▶−1,x^2▶−x,x)enter

Example 8.24

Degree

1. Type polyDegree(x³+x, x).

   polydegree(x^3▶+x,x)enter

Example 8.25

Coefficients

1. Type polyCoeffs(x²+3x+2, x).

   polycoeffs(x^2▶+3x+2,x)enter