Write in Standard Form (x^2+3x+2)(x^2-3x-4)

Solution:

Step 1:

To express a polynomial in standard form, first expand the product and then order the resulting terms from highest to lowest degree, following the general form $a{x}^n+b{x}^{n-1}+\dots +c$.

Step 2:

Multiply out the binomials $(x^2+3x+2)(x^2-3x-4)$ by distributing each term in the first binomial across each term in the second binomial, resulting in $x^2\cdot x^2+x^2\cdot (-3x)+x^2\cdot (-4)+3x\cdot x^2+3x\cdot (-3x)+3x\cdot (-4)+2\cdot x^2+2\cdot (-3x)+2\cdot (-4)$.

Step 3:

Simplify the expression by combining like terms.

Step 3.1:

Identify and combine like terms in the expanded expression.

Step 3.1.1:

Rearrange the terms involving $x^2\cdot (-3x)$ and $3x\cdot x^2$ to $x^2\cdot x^2-3x^3+x^2\cdot (-4)+3x^3+3x\cdot (-3x)+3x\cdot (-4)+2\cdot x^2+2\cdot (-3x)+2\cdot (-4)$.

Step 3.1.2:

Cancel out $-3x^3$ and $3x^3$ to get $x^2\cdot x^2+x^2\cdot (-4)+3x\cdot (-3x)+3x\cdot (-4)+2\cdot x^2+2\cdot (-3x)+2\cdot (-4)$.

Step 3.1.3:

Combine the terms $x^2\cdot x^2$ and $x^2\cdot (-4)$.

Step 3.2:

Simplify each term by performing the multiplication.

Step 3.2.1:

Apply the exponent rule $a^m\cdot a^n=a^{m+n}$ to multiply $x^2$ by $x^2$, resulting in $x^4$.

Step 3.2.2:

Reposition $-4$ to the left of $x^2$ to maintain standard form.

Step 3.2.3:

Use the commutative property of multiplication to rewrite terms.

Step 3.2.4:

Multiply $x$ by $x$ by adding their exponents.

Step 3.2.5:

Multiply $3$ by $-3$ to get $-9$.

Step 3.2.6:

Multiply $-4$ by $3$ to get $-12$.

Step 3.2.7:

Multiply $-3$ by $2$ to get $-6$.

Step 3.2.8:

Multiply $2$ by $-4$ to get $-8$.

Step 3.3:

Combine like terms to simplify the expression.

Step 3.3.1:

Combine $-4x^2$ and $-9x^2$.

Step 3.3.2:

Add $-13x^2$ and $2x^2$.

Step 3.3.3:

Combine $-12x$ and $-6x$.

The final standard form of the polynomial is $x^4-11x^2-18x-8$.

Knowledge Notes:

1. Standard Form of a Polynomial: A polynomial is in standard form when its terms are ordered from highest to lowest degree. The general form is $a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x+a_0$, where $a_n$ is the leading coefficient and $a_0$ is the constant term.

2. Multiplying Polynomials: To multiply polynomials, distribute each term in the first polynomial by each term in the second polynomial. This is often done by using the FOIL method for binomials or by applying the distributive property more generally.

3. Combining Like Terms: Terms in an expression that have the same variable raised to the same power are called like terms. These can be combined by adding or subtracting their coefficients.

4. Exponent Rules: When multiplying terms with the same base, add the exponents (power rule: $a^m\cdot a^n=a^{m+n}$). When raising a power to a power, multiply the exponents (power of a power rule: $(a^m{)}^n=a^{m\cdot n}$).

5. Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not affect the product ($a\cdot b=b\cdot a$).

6. Simplifying Expressions: Simplifying an algebraic expression involves performing all possible operations, including distributing and combining like terms, to write the expression in its simplest form.