Find the Product (12x^2+7y^3)(4x^2+7y^3)
The question provided is asking you to perform a mathematical operation called multiplication on two algebraic expressions which are binomials (expressions containing two terms). The first binomial is (12x^2+7y^3) and the second is (4x^2+7y^3). The operation requires you to apply the distributive property, often referred to as FOIL (First, Outside, Inside, Last) when multiplying binomials, to find the product of these two expressions. You will need to multiply each term in the first binomial by each term in the second binomial and then combine like terms if necessary to simplify the resulting polynomial expression.
$\left(\right. 12 x^{2} + 7 y^{3} \left.\right) \left(\right. 4 x^{2} + 7 y^{3} \left.\right)$
Answer
Solution:
Step 1: Expand the Expression
Use the FOIL (First, Outer, Inner, Last) method to expand $(12x^2 + 7y^3)(4x^2 + 7y^3)$.
Step 1.1: Distribute the Terms
Distribute the terms as follows: $12x^2(4x^2 + 7y^3) + 7y^3(4x^2 + 7y^3)$.
Step 1.2: Continue Distribution
Further distribute to get $12x^2(4x^2) + 12x^2(7y^3) + 7y^3(4x^2) + 7y^3(7y^3)$.
Step 1.3: Complete Distribution
Complete the distribution to obtain $12x^2 \cdot 4x^2 + 12x^2 \cdot 7y^3 + 7y^3 \cdot 4x^2 + 7y^3 \cdot 7y^3$.
Step 2: Simplify and Combine Like Terms
Step 2.1: Simplify Each Term
Step 2.1.1: Use Commutative Property
Use the commutative property to rewrite as $12 \cdot 4x^2 \cdot x^2 + 12x^2 \cdot 7y^3 + 7y^3 \cdot 4x^2 + 7y^3 \cdot 7y^3$.
Step 2.1.2: Multiply Powers of x
Apply the power rule for multiplication of exponents: $x^m \cdot x^n = x^{m+n}$.
Step 2.1.2.1: Rearrange Terms
Rearrange to $12 \cdot 4(x^2 \cdot x^2) + 12x^2 \cdot 7y^3 + 7y^3 \cdot 4x^2 + 7y^3 \cdot 7y^3$.
Step 2.1.2.2: Combine Exponents
Combine the exponents of x to get $12 \cdot 4x^{2+2} + 12x^2 \cdot 7y^3 + 7y^3 \cdot 4x^2 + 7y^3 \cdot 7y^3$.
Step 2.1.2.3: Add Exponents
Add the exponents to simplify to $12 \cdot 4x^4 + 12x^2 \cdot 7y^3 + 7y^3 \cdot 4x^2 + 7y^3 \cdot 7y^3$.
Step 2.1.3: Multiply Constants
Multiply the constants: $48x^4 + 12x^2 \cdot 7y^3 + 7y^3 \cdot 4x^2 + 7y^3 \cdot 7y^3$.
Step 2.1.4: Use Commutative Property Again
Use the commutative property to rewrite as $48x^4 + 12 \cdot 7x^2y^3 + 7y^3 \cdot 4x^2 + 7y^3 \cdot 7y^3$.
Step 2.1.5: Multiply More Constants
Multiply constants to get $48x^4 + 84x^2y^3 + 7y^3 \cdot 4x^2 + 7y^3 \cdot 7y^3$.
Step 2.1.6: Commutative Property Once More
Use the commutative property again to rewrite as $48x^4 + 84x^2y^3 + 7 \cdot 4y^3x^2 + 7y^3 \cdot 7y^3$.
Step 2.1.7: Multiply Remaining Constants
Multiply the remaining constants to obtain $48x^4 + 84x^2y^3 + 28y^3x^2 + 7y^3 \cdot 7y^3$.
Step 2.1.8: Final Commutative Property Use
Use the commutative property for the last time to rewrite as $48x^4 + 84x^2y^3 + 28y^3x^2 + 7 \cdot 7y^3y^3$.
Step 2.1.9: Multiply Powers of y
Apply the power rule for y: $y^m \cdot y^n = y^{m+n}$.
Step 2.1.9.1: Rearrange y Terms
Rearrange to get $48x^4 + 84x^2y^3 + 28y^3x^2 + 7 \cdot 7(y^3 \cdot y^3)$.
Step 2.1.9.2: Combine y Exponents
Combine the exponents of y to get $48x^4 + 84x^2y^3 + 28y^3x^2 + 7 \cdot 7y^{3+3}$.
Step 2.1.9.3: Add y Exponents
Add the exponents to simplify to $48x^4 + 84x^2y^3 + 28y^3x^2 + 7 \cdot 7y^6$.
Step 2.1.10: Multiply Final Constants
Multiply the final constants to get $48x^4 + 84x^2y^3 + 28y^3x^2 + 49y^6$.
Step 2.2: Combine Like Terms
Step 2.2.1: Rearrange Like Terms
Rearrange like terms to get $48x^4 + 84x^2y^3 + 28x^2y^3 + 49y^6$.
Step 2.2.2: Combine Like Terms
Combine $84x^2y^3$ and $28x^2y^3$ to get $48x^4 + 112x^2y^3 + 49y^6$.
The final result is $48x^4 + 112x^2y^3 + 49y^6$.
Knowledge Notes:
FOIL Method: A technique for expanding binomials, where you multiply the First, Outer, Inner, and Last terms of the binomial expressions.
Distributive Property: A property that allows you to multiply a sum by multiplying each addend separately and then add the products.
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not change the product, i.e., $a \cdot b = b \cdot a$.
Power Rule for Exponents: When multiplying like bases, you add the exponents, i.e., $a^m \cdot a^n = a^{m+n}$.
Combining Like Terms: When simplifying expressions, terms with the same variable factors can be combined by adding or subtracting the coefficients.
