1.1
Algebraic Expressions and Factoring
Expanding Polynomials
An algebraic expression is a polynomial of the form \( a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0 \) where \(a_0\), \(a_1\), \(\ldots\), \(a_n\) are real numbers and \(n\) is non-negative. If \(a_n\neq0\), the degree of the polynomial is \(n\).
The FOIL method is a way to expand a factored polynomal.
- Multiply (expand) this polynomial: \( (2x+1)(3x-5) \) \[\begin{align} &(2x*3x)+(2x*-5)+(1*3x)+(1*-5) \\ &6x^2-10x+3x-5 \\ &6x^2-7x-5 \end{align}\]
Special Product Formulas
These special product formulas can be used to expand a factored polynomial that follows one of the specified patterns.
| \( (A-B)(A+B) \) | \( A^2-B^2 \) |
|---|---|
| \( (A+B)^2 \) | \( A^2+2AB+B^2 \) |
| \( (A-B)^2 \) | \( A^2-2AB+B^2 \) |
| \( (A+B)^3 \) | \( A^3+3A^2B+3AB^2+B^2 \) |
| \( (A-B)^3 \) | \( A^3-3A^2B+3AB^2-B^3 \) |
- Expand this expression: \( (3x+5)^2 \) \[\begin{align} A&=3x \\ B&=5 \end{align}\] \[\begin{align} &(3x)^2 + 2(3x)(5) + 5^2 \\ &9x^2+30x+25 \end{align}\]
- Expand this expression: \( (x^2-2)^3 \) \[\begin{align} A&=x^2 \\ B&=2 \end{align}\] \[\begin{align} &(x^2)^3-3(x^2)^2(2)+3(x^2)(2)^2-2^3 \\ &x^6-6x^4+12x^2+8 \end{align}\]
Special Factoring Formulas
These special factoring formulas can be used to factor an expanded polynomial that follows one of the specified patterns.
| \( A^2-B^2 \) | \( (A-B)(A+B) \) |
|---|---|
| \( A^2+2AB+B^2 \) | \( (A+B)^2 \) |
| \( A^2-2AB+B^2 \) | \( (A-B)^2 \) |
| \( A^3-B^3 \) | \( (A-B)(A^2+BA+B^2) \) |
| \( A^3+B^3 \) | \( (A+B)(A^2-BA+B^2) \) |
- Factor this expression: \( 4x^2-25 \) \[\begin{align} &(2x^2)-(5)^2 \\ &(2x-5)(2x+5) \end{align}\]
- Factor this expression: \( 27x^3-1 \) \[\begin{align} &(3x)^3-(1)^3 \\ &(3x-1)[(3x)^2+1(3x)+1^2] \\ &(3x-1)(9x^2+3x+1) \end{align}\]
Exponent Laws
When working with polynomials, you may need to recall how to combine, expand, and utilize exponents. These exponent laws determine the functionality of exponents and their bases.
| \( a^m a^n \) | \( a^{m+n} \) |
|---|---|
| \( (a^m)^n \) | \( a^{mn} \) |
| \( (ab)^n \) | \( a^n b^n \) |
| \( \frac{a^m}{a^n} \) | \( a^{m-n} \) |
| \( (\frac{a}{b})^n \) | \( \frac{a^n}{b^n} \) |
| \( (\frac{a}{b})^{-n} \) | \( (\frac{b}{a})^n \) |
| \( \frac{a^{-n}}{b^{-m}} \) | \( \frac{b^m}{a^n} \) |