Simplify (x^3+2x^2)(3x^2-1)
The question is asking for the simplification of a mathematical expression which is the product of two binomials. The first binomial is \( x^3 + 2x^2 \) and the second binomial is \( 3x^2 - 1 \). To simplify the expression, one would typically use the distributive property (also known as the FOIL method - which stands for First, Outer, Inner, Last) to multiply each term in the first binomial by each term in the second binomial, and then combine like terms if necessary. The goal is to express the product as a single polynomial in its simplest form.
$\left(\right. x^{3} + 2 x^{2} \left.\right) \left(\right. 3 x^{2} - 1 \left.\right)$
Answer
Solution:
Step 1: Distribute each term in the first polynomial to each term in the second polynomial.
- Multiply $x^3$ by $(3x^2 - 1)$ and $2x^2$ by $(3x^2 - 1)$.
Step 1.1: Apply the distributive property.
- $x^3(3x^2 - 1) + 2x^2(3x^2 - 1)$
Step 1.2: Distribute $x^3$ across $(3x^2 - 1)$.
- $x^3 \cdot 3x^2 + x^3 \cdot (-1) + 2x^2(3x^2 - 1)$
Step 1.3: Distribute $2x^2$ across $(3x^2 - 1)$.
- $x^3 \cdot 3x^2 + x^3 \cdot (-1) + 2x^2 \cdot 3x^2 + 2x^2 \cdot (-1)$
Step 2: Combine like terms.
Step 2.1: Simplify each term individually.
Step 2.1.1: Use the commutative property to reorder multiplication.
- $3x^5 - x^3 + 6x^4 - 2x^2$
Step 2.1.2: Combine the exponents where multiplication occurs.
- Use the power rule $a^m \cdot a^n = a^{m+n}$.
Step 2.1.3: Simplify the expression by combining like terms.
- The final simplified expression is $3x^5 + 6x^4 - x^3 - 2x^2$.
Knowledge Notes:
The problem-solving process involves simplifying a polynomial expression by multiplying two binomials. Here are the relevant knowledge points:
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to multiply a single term by each term in a binomial or polynomial.
FOIL Method: This is a specific case of the distributive property used for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to the terms in each binomial that are multiplied together.
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not change the product, i.e., $ab = ba$.
Power Rule: When multiplying like bases, you add the exponents, as in $a^m \cdot a^n = a^{m+n}$.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power.
Simplifying Expressions: The process of reducing an expression to its simplest form by performing all possible operations and combining like terms.
In the given solution, these principles are applied step by step to expand and simplify the given polynomial expression. The solution is presented in a structured format, with each step building upon the previous one to reach the final simplified form.
