Understanding Polynomials

Polynomials are expressions that contain variables with whole number exponents. They come in different forms based on how many terms they have:

• Monomials have just one term (like 2x)
• Binomials contain two terms $like 2x⁴ + x$
• Trinomials have three terms $like 4x³ - 4x² + 6$
• Expressions with more terms are simply called polynomials

The degree of a polynomial is the highest exponent in any term. For example, in x³ + x² - x + 2, the degree is 3.

💡 When identifying polynomials, always look at the number of terms and the highest exponent to determine both the type and degree.

Like terms are terms with the same variable and the same exponent. You can combine like terms by adding or subtracting their coefficients. For example:

• 4x + 2x = 6x (same variable, same exponent)
• 3y + 7y = 10y (like terms)
• 8y - 5p $cannot be combined - different variables$

Exponents and Their Properties

Exponents tell us how many times to multiply a base by itself. Understanding how they work is crucial for algebra success!

The basic formula is: base^exponent = result

• 5⁴ = 625 (5 multiplied by itself 4 times)
• 3³ = 27 (3 × 3 × 3)

When calculating expressions with exponents:

1. Solve the exponent operation first
2. Then perform multiplication/division

For example:

• 2(3)³ = 2(27) = 54
• -5(2)³ = -5(8) = -40

Exponential growth appears in real-world applications like compound interest. If you invest $500 at 3.5$1,403.40!

💡 Remember PEMDAS when working with exponents: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

When dividing expressions with exponents, you can subtract the exponents when the bases are the same:

• 2⁴/2² = 2² = 4
• 5³/1 = 5³ = 125

Adding and Subtracting Polynomials

Adding and subtracting polynomials is all about identifying and combining like terms. Like terms have the same variables raised to the same powers.

To add or subtract polynomials:

1. Group like terms together
2. Combine their coefficients
3. Keep the variables and their exponents the same

Examples:

• 12p³ + 11p³ + 8p³ = 31p³ (all like terms)
• 5x² - 6 - 3x + 8 = 5x² - 3x + 2 (grouped like terms)
• $4m² + 5$ + $m² + m + 6$ = 5m² + m + 11 (combined like terms)

When subtracting polynomials, be careful with signs:

• $x³ + 4y$ - (2x³) = x³ + 4y - 2x³ = -x³ + 4y (distributed the negative)
• $-10x² - 3x + 7$ - $x² + 1$ = -10x² - 3x + 7 - x² - 1 = -11x² - 3x + 6

💡 A quick check: When adding or subtracting, the variables and their exponents never change—only the coefficients (numbers) in front of them change.

Practice identifying like terms first, then combining them will become second nature!

Multiplying Polynomials

When you multiply polynomials, you'll need to apply the distributive property. Let's see how it works:

Multiplying Monomials:

1. Multiply the coefficients together
2. Add the exponents of like variables

Examples:

• (6y³)(3y⁵) = 18y⁸ (multiplied coefficients, added exponents)
• (8mn²)(9m²n) = 72m³n³ (added exponents for each variable)
• (3x³)(6x²) = 18x⁵ $3 × 6 = 18, x³ × x² = x⁵$

Multiplying a Monomial by a Polynomial: Use the distributive property - multiply each term in the polynomial by the monomial.

Examples:

• 4$3x² + 4x - 8$ = 12x² + 16x - 32
• 6pq²$2p² - q$ = 12p³q² - 6pq³

💡 When multiplying expressions with variables, remember to: (1) multiply the coefficients, and (2) add the exponents of the same variable.

This pattern applies to more complex expressions too. Just take it one step at a time, and you'll master this important algebra skill!

More Polynomial Multiplication

Let's explore some more examples of multiplying polynomials:

When multiplying a monomial by a binomial or trinomial:

1. Distribute the monomial to each term
2. Multiply coefficients and add exponents for variables

Examples:

• 2x²$-3x²y + 2x²y²$ = -6x⁴y + 4x⁴y²
• 2$4x² + x + 3$ = 8x² + 2x + 6
• 3ab$5a² + b$ = 15a³b + 3ab²

Checking your work: You can verify your polynomial multiplications by plugging in simple values for the variables. If both the original expression and your answer give the same result, your multiplication is probably correct!

💡 A common mistake is forgetting to distribute the monomial to ALL terms in the polynomial. Make sure to multiply each term separately.

Remember that when multiplying terms with the same variable, you add the exponents:

• x² × x = x³
• a² × a² = a⁴
• y × y² = y³

This fundamental skill will help you tackle more complex algebraic problems in the future!

The FOIL Method for Binomial Multiplication

When multiplying two binomials, the FOIL method gives us a systematic approach:

First: Multiply the first terms in each binomial Outer: Multiply the outer terms Inner: Multiply the inner terms Last: Multiply the last terms

Example: $x + 3$$x + 2$

• F: x × x = x²
• O: x × 2 = 2x
• I: 3 × x = 3x
• L: 3 × 2 = 6

Combine like terms: x² + 2x + 3x + 6 = x² + 5x + 6

More examples:

• $x + 1$$x - 1$ = x² - 1x + 1x - 1 = x² - 1
• $x - 1$$x + 2$ = x² + 2x - 1x - 2 = x² + x - 2
• $b - 3$$b - 3$ = b² - 3b - 3b + 9 = b² - 6b + 9

💡 The FOIL method only works for multiplying two binomials. For larger expressions, you'll need to use the distributive property more extensively.

Another approach for multiplying polynomials is using an area model (box method), which we'll see later. Both techniques lead to the same result!

Practice with Polynomials

Let's practice what we've learned about polynomial operations:

Adding like terms:

• 3x² + 6x² = 9x² (combine coefficients, keep the variable and exponent)
• 4x³ + x³ = 5x³ (add coefficients of like terms)

Multiplying polynomials:

• 2$2x² + 3x - 1$ = 4x² + 6x - 2 (distribute the 2)
• 3a²$ab + a$ = 3a²(ab) + 3a²(a) = 3a³b + 3a³ (distribute and add exponents)
• 6xy²$xy + 3x²y$ = 6x²y³ + 18x³y³ (distribute and add exponents)

When multiplying terms with the same variable base:

1. Keep the base
2. Add the exponents
3. Multiply the coefficients

💡 Pay attention to the exponents in each term. Remember that any variable without a written exponent has an implied exponent of 1 $for example, x = x¹$.

These skills form the foundation for more advanced topics in algebra, like factoring polynomials and solving quadratic equations.

Binomial Multiplication and Special Products

When multiplying binomials, we can use either the FOIL method or distribute one binomial to each term in the other.

For example, $a + 3$$a - 3$:

• a$a - 3$ + 3$a - 3$
• a² - 3a + 3a - 9
• a² - 9

Let's practice with more complex examples:

$8m² - n$$m² - 3n$

• F: 8m² × m² = 8m⁴
• O: 8m² × $-3n$ = -24m²n
• I: $-n$ × m² = -m²n
• L: $-n$ × $-3n$ = 3n²
• Result: 8m⁴ - 24m²n - m²n + 3n² = 8m⁴ - 25m²n + 3n²

$2a - b²$$a + 4b²$

• F: 2a × a = 2a²
• O: 2a × 4b² = 8ab²
• I: $-b²$ × a = -ab²
• L: $-b²$ × 4b² = -4b⁴
• Result: 2a² + 8ab² - ab² - 4b⁴ = 2a² + 7ab² - 4b⁴

💡 When multiplying binomials with more complex terms, be extra careful with signs. A negative times a negative gives a positive result!

The FOIL method is a great tool, but as you become more comfortable, you might find that distributing works better for certain problems. Use whichever method makes more sense to you!

Special Products and Perfect Square Trinomials

Sometimes, certain patterns appear when multiplying binomials. These are called special products and recognizing them can save you time.

The most common special products are:

1. Perfect Square Trinomial (First Pattern): $a + b$² = a² + 2ab + b²

2. Perfect Square Trinomial (Second Pattern): $a - b$² = a² - 2ab + b²

3. Difference of Squares: $a + b$$a - b$ = a² - b²

For example:

• $x + 3$² = x² + 2(x)(3) + 3² = x² + 6x + 9
• $2x - 5$² = (2x)² - 2(2x)(5) + 5² = 4x² - 20x + 25

The box method is another way to multiply polynomials:

1. Create a box with rows and columns labeled with each term
2. Multiply the terms at each intersection
3. Add all the products inside the box

💡 Perfect square trinomials always follow the pattern a² + 2ab + b² or a² - 2ab + b². If you see a trinomial in this form, it can be factored as $a + b$² or $a - b$².

Understanding these patterns will help you both multiply and factor polynomials more efficiently!

More Special Products Practice

Let's practice recognizing and applying special product formulas:

Perfect Square Trinomial Formula: $a + b$² = a² + 2ab + b²

Examples:

• $4s + 3t$² = (4s)² + 2(4s)(3t) + (3t)² = 16s² + 24st + 9t²
• $5 + m²$² = 5² + 2(5)(m²) + (m²)² = 25 + 10m² + m⁴
• $x + 6$² = x² + 2(x)(6) + 6² = x² + 12x + 36
• $5a + b$² = (5a)² + 2(5a)(b) + b² = 25a² + 10ab + b²
• $1 + c³$² = 1² + 2(1)(c³) + (c³)² = 1 + 2c³ + c⁶

Perfect Square Trinomial (negative version): $a - b$² = a² - 2ab + b²

Example:

• $x - 4$² = x² - 2(x)(4) + 4² = x² - 8x + 16

💡 When working with perfect square trinomials, identify what "a" and "b" are first. Then substitute into the formula. This approach is much faster than using FOIL for these special cases.

Learning these patterns will make both multiplication and factoring much easier as you progress in algebra!