Solving Multi-Step Algebra Equations Notes

\begin{definition}[Polynomial] A polynomial is an expression of the form:

$$\ a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\ $$

where:

• \$a_n, a_{n-1}, \ldots, a_1, a_0\$ are coefficients (real numbers).
• \$x\$ is a variable.
• \$n\$ is a non-negative integer called the degree of the polynomial. \end{definition}

\begin{callout}[type=note] The degree of a polynomial is the highest power of the variable \$x\$ with a non-zero coefficient. \end{callout}

Multiplying Polynomials

Multiplying Binomials

A binomial is a polynomial with two terms, such as \$(x + 3)\$ or \$(2x - 5)\$.

To multiply two binomials, we use the FOIL method:

1. First: Multiply the first terms of each binomial.
2. Outer: Multiply the outer terms.
3. Inner: Multiply the inner terms.
4. Last: Multiply the last terms.

\begin{callout}[type=example] Multiply \$(x + 3)(2x - 4)\$.

1. First: \$x \cdot 2x = 2x^2\$
2. Outer: \$x \cdot (-4) = -4x\$
3. Inner: \$3 \cdot 2x = 6x\$
4. Last: \$3 \cdot (-4) = -12\$

Combine the results:

$$\ 2x^2 - 4x + 6x - 12 = 2x^2 + 2x - 12\ $$ \end{callout}

Multiplying Polynomials of Higher Degree

When multiplying polynomials with more than two terms, the FOIL method does not apply. Instead, we use the distributive property:

1. Multiply each term in the first polynomial by each term in the second polynomial.
2. Combine like terms.

\begin{callout}[type=example] Multiply \$(x + 3)(2x^2 - 4x + 5)\$.

2. Multiply \$x\$ by each term in \$2x^2 - 4x + 5\$:
   • \$x \cdot 2x^2 = 2x^3\$
   • \$x \cdot (-4x) = -4x^2\$
   • \$x \cdot 5 = 5x\$
4. Multiply \$3\$ by each term in \$2x^2 - 4x + 5\$:
   • \$3 \cdot 2x^2 = 6x^2\$
   • \$3 \cdot (-4x) = -12x\$
   • \$3 \cdot 5 = 15\$
6. Combine the results:

$$\ 2x^3 - 4x^2 + 5x + 6x^2 - 12x + 15 = 2x^3 + 2x^2 - 7x + 15\ $$ \end{callout}

Multiplication Patterns

Perfect Square Trinomials

When a binomial is squared, it follows a specific pattern:

$$\ (x + a)^2 = x^2 + 2ax + a^2\ $$

\begin{callout}[type=example] Simplify \$(x + 5)^2\$.

1. The coefficient of \$x\$ is \$2 \cdot 5 = 10\$.
2. The constant term is \$5^2 = 25\$.

So, \$(x + 5)^2 = x^2 + 10x + 25\$. \end{callout}

Difference of Squares

When two binomials differ only by the sign between their terms, the product is a difference of squares:

$$\ (x - a)(x + a) = x^2 - a^2\ $$

\begin{callout}[type=example] Simplify \$(x - 3)(x + 3)\$.

The answer is \$x^2 - 3^2 = x^2 - 9\$. \end{callout}

Dividing Polynomials

Long Division

Dividing polynomials is similar to long division for numbers.

\begin{callout}[type=example] Divide \$2x^3 + x^2 - 11x + 12\$ by \$x + 3\$.

1. Determine what to multiply \$x\$ by to get \$2x^3\$: \$2x^2\$.
2. Multiply \$2x^2\$ by \$x + 3\$: \$2x^3 + 6x^2\$.
3. Subtract from the original polynomial: \$2x^3 + x^2 - 11x + 12 - (2x^3 + 6x^2) = -5x^2 - 11x\$.
4. Repeat the process for \$-5x^2\$: Multiply by \$-5x\$.
5. Continue until all terms are divided. \end{callout}

\begin{callout}[type=note] If there is a remainder after division, it can be expressed as a fraction:

$$\ \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}\ $$ \end{callout}

\begin{callout}[type=self_review]

1. Multiply \$(2x - 3)(x^2 + 4x + 5)\$.
2. Divide \$3x^3 - 2x^2 + 4x - 1\$ by \$x - 1\$. \end{callout}

\begin{callout}[type=tok] How do patterns like the difference of squares or perfect square trinomials help us understand the structure of algebraic expressions? Are these patterns discovered or invented? \end{callout}