Lesson 2: Simplifying Algebraic Expressions I
MA158 - Spring 2026
Review of Polynomials
Definition: Polynomial
A polynomial is of the form
\[a_n x^n + a_{n-1} x^{n-1} + \ldots + a_2 x^2 + a_1 x + a_0 x^0\]for a whole number \(n \geq 0\), where the exponents are decreasing. This is called the standard form of a polynomial.
The terms are labeled as follows:
- Degree: \(n\)
- Leading term: \(a_n x^n\) (the term with the highest degree)
- Quadratic term: \(a_2 x^2\)
- Linear term: \(a_1 x\)
- Constant term: \(a_0\)
Key Polynomial Terminology
Coefficients: The values \(a_i\) where \(i\) is a positive integer.
Degree: The highest power of the variable with a non-zero coefficient.
Special types of polynomials based on number of terms:
- Monomial: one term
- Binomial: two terms
- Trinomial: three terms
Different degrees:
- Constant (degree 0)
- Linear (degree 1)
- Quadratic (degree 2)
- Cubic (degree 3)
Important Facts
Fact 1: The sum or difference of polynomials is also a polynomial.
Fact 2: The product of polynomials is also a polynomial.
Operations with Polynomials
Addition of Polynomials
Example 1: Sum the following polynomials: \(x^2 + 2x + 1\) and \(x^3 + x + 1\)
Solution:
\[\begin{align} &(x^2 + 2x + 1) + (x^3 + x + 1)\\ &= x^3 + x^2 + 2x + x + 1 + 1\\ &= x^3 + x^2 + 3x + 2 \end{align}\]Alternatively, we can align like terms vertically:
\[\begin{array}{rrrrr} & & x^2 & + 2x & + 1\\ + & x^3 & & + x & + 1\\ \hline & x^3 & + x^2 & + 3x & + 2 \end{array}\]Multiplication of Polynomials
Example 2: Find the product of \((x^2 + 1)\) and \((2x - 5)\)
Way 1: Distribute
\[\begin{align} (x^2 + 1)(2x - 5) &= x^2(2x - 5) + 1(2x - 5)\\ &= 2x^3 - 5x^2 + 2x - 5 \end{align}\]Way 2: Table Method
We create a table with the terms of the first polynomial along the top and the terms of the second polynomial along the left side:
| \(x^2 + 1\) | |||
|---|---|---|---|
| \(x^2\) | \(+1\) | ||
| \(2x + 5\) | \(2x\) | \(2x^3\) | \(2x\) |
| \(+5\) | \(5x^2\) | \(5\) | |
Answer: \(2x^3 + 5x^2 + 2x + 5\)
The process we just did is called expanding (or multiplying out, or distributing, etc.). Expanding can be one way of simplifying.
Example 3: Simplify \((x - y)(x^2 + xy + y^2) = x^3 - xy + x^2y - xy^2 + xy^2 - y^3\)
Using a table to organize the multiplication:
| \(x^2\) | \(xy\) | \(y^2\) | |
|---|---|---|---|
| \(x\) | \(x^3\) | \(x^2y\) | \(xy^2\) |
| \(-y\) | \(-x^2y\) | \(-xy^2\) | \(-y^3\) |
Combining like terms (the middle terms eliminate):
\[= x^3 - y^3\]Greatest Common Factor (GCF)
Definition: Greatest Common Factor
The greatest common factor of the terms of a polynomial is the "largest" (largest coefficient, highest degree) monomial that divides all the terms.
We can also find GCFs of expressions not in standard form.
Example 4: Find the GCF of the following:
(a) \(36x^2y^2 - 18x^2y^3 + 30x^5y\)
We find the GCF of the coefficients: \(\text{GCF}(36, 18, 30) = 6\)
We find the GCF of the variable parts: \(\text{GCF}(x^2y^2, x^2y^3, x^5y) = x^2y\)
\[= 6x^2y[6xy - 3y^2 + 5x^3]\]GCF: \(6x^2y\)
(b) (non-standard form) \(4(x+2)^3(x+6) - 20(x+2)^2(x+6)^3\)
We identify the common factors:
- Numerical GCF: \(4\)
- Common factor \((x+2)\): appears as \((x+2)^3\) and \((x+2)^2\), so we take \((x+2)^2\)
- Common factor \((x+6)\): appears as \((x+6)\) and \((x+6)^3\), so we take \((x+6)\)
GCF: \(4(x+2)^2(x+6)\)
Factoring Polynomials
Definition: Factoring
Factoring is the process of breaking up a polynomial into the product of two or more polynomials of lower degree.
(Factoring is the opposite process of expanding.)
The relationship between expanding and factoring:
\[ (x+1)(2x+5) \xrightarrow{\text{expand}} 2x^3 + 5x^2 + 2x + 5 \xrightarrow{\text{factor}} (x+1)(2x+5) \]Important: Before trying to factor, find GCF and factor that first.
Factoring by Grouping
Sometimes we can group the terms of a polynomial to factor it. To be successful, each group must have the same number of terms.
Example 5: Factor \(x^6 + 3x^5 + 2x^2 + 6x\)
We group the terms in pairs and factor out the GCF from each group:
\[\begin{align} &= x^5(x+3) + 2x(x+3)\\ &= (x+3)(x^5 + 2x) \end{align}\]We can factor further:
\[\begin{align} &= (x+3)(x^5 + 2x)\\ &= (x+3)(x)(x^4 + 2) \end{align}\]Example 6: Factor \(2x^2 - 3x + 2x - 3\)
We group the terms in pairs and factor out the GCF from each group:
\[\begin{align} &= (2x^2 - 3x) + (2x - 3)\\ &= x(2x - 3) + 1(2x - 3)\\ &= (x+1)(2x-3) \end{align}\]Factoring Quadratics
Note: Sometimes factoring is possible, but sometimes it is not. If not, the polynomial is irreducible.
Factoring Quadratics with No Linear Term
We can use the fact that \(a^2 - b^2 = (a+b)(a-b)\)
This is called the difference of squares formula.
Example 7: Factor \(9x^2 - 25\)
Example 8: Factor \(16y^4 - 121\)
Factoring Quadratics with a Linear Term
Consider quadratics of the form \(ax^2 + bx + c\).
The method is the ac-method.
Example 9: Factor \(x^2 - 5x - 14\)
Given: \(a = 1\), \(b = -5\), \(c = -14\)
We need to find two numbers whose product is \(ac = 1 \cdot (-14) = -14\) and whose sum is \(b = -5\).
The numbers are \(-7\) and \(2\) because:
- \((-7) \times 2 = -14\) ✓
- \((-7) + 2 = -5\) ✓
We rewrite the middle term using these numbers:
\[\begin{align} x^2 - 5x - 14 &= x^2 + 2x - 7x - 14\\ &= x(x+2) - 7(x+2)\\ &= (x-7)(x+2) \end{align}\]Example 10: Factor \(6t^2 - 11t - 35\)
Given: \(a = 6\), \(b = -11\), \(c = -35\)
We calculate \(ac = 6 \times (-35) = -210\)
We need two numbers whose product is \(-210\) and whose sum is \(-11\).
After checking factor pairs of 210, we find that \(-21\) and \(10\) work because:
- \((-21) \times 10 = -210\) ✓
- \((-21) + 10 = -11\) ✓
Rewrite the middle term using \(-21\) and \(10\):
\[\begin{align} 6t^2 - 11t - 35 &= 6t^2 - 21t + 10t - 35 \end{align}\]Factor by grouping:
\[\begin{align} &= (6t^2 - 21t) + (10t - 35)\\ &= 3t(2t - 7) + 5(2t - 7)\\ &= (2t - 7)(3t + 5) \end{align}\]