Problem 87 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ \left(3 x^{2}-5\right)\left(2 x^{2}+1\right) $$

Step-by-Step Solution

Answer

Question: Multiply the two expressions (3x² - 5) and (2x² + 1) and simplify the result. Answer: The product of the two expressions is 6x⁴ - 7x² - 5.

Step 1: Multiply the First Terms

First, we need to multiply the first terms from both expressions: \((3x^2)(2x^2)\). Using the rules of exponents, we get: $$ (3x^2)(2x^2) = 6x^{2+2} = 6x^4 $$

Step 2: Multiply the Inner and Outer Terms

Now, let's multiply the inner terms and the outer terms: Inner terms: \((3x^2)(1)\): $$ (3x^2) = 3x^2 $$ Outer terms: \((-5)(2x^2)\): $$ (-5)(2x^2) = -10x^2 $$

Step 3: Multiply the Last Terms

Finally, we need to multiply the last terms from both expressions: \((-5)(1)\) $$ (-5)(1) = -5 $$

Step 4: Combine like terms

Now, let's combine all the terms we have obtained after multiplication: $$ 6x^4 + 3x^2 - 10x^2 - 5 $$ Combine the two middle terms since they are like and have the same power of x: $$ 6x^4 - 7x^2 - 5 $$

Step 5: Write the final result

After combining all the like terms, we get the simplified and final expression as follows: $$ (3x^2 - 5)(2x^2 + 1) = 6x^4 - 7x^2 - 5 $$

Key Concepts

Like Terms

Like terms are essential when simplifying algebraic expressions, especially during polynomial multiplication. They are terms in an expression that have the same variable raised to the same power. For example, in the expression \(6x^4 + 3x^2 - 10x^2 - 5\), the terms \(3x^2\) and \(-10x^2\) are considered like terms since they both involve \(x\) raised to the second power.
Why do we combine like terms? Combining them allows us to simplify expressions, making them easier to understand and use in further calculations.
Here's how you can spot and combine like terms:

• Identify terms with the same variable and exponent.
• Add or subtract their coefficients (numerical parts).

In our example, combining \(3x^2\) and \(-10x^2\) gives you \(-7x^2\).
Remember, only modify the coefficients and not the variable parts when working with like terms.

Exponent Rules

Understanding exponent rules is crucial to simplify expressions and multiply polynomials correctly. Exponent rules help manage expressions where the same base is raised to different powers. Let's go through the most relevant rule applied in our exercise:

• Product of Powers Rule: This rule states that when you multiply two powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\). In our exercise, multiplying \(3x^2\) with \(2x^2\) uses this rule: \((3)(2)(x^2 \times x^2) = 6x^{2+2}\), resulting in \(6x^4\).

It's important to apply these rules consistently to ensure accuracy. Remember, even small errors in applying exponent rules can lead to incorrect results!
Keep practicing applying these rules to become more comfortable with polynomial problems.

Distributive Property

The distributive property is a key algebraic property useful for polynomial multiplication. It states \(a(b + c) = ab + ac\). This property allows us to multiply single terms and larger polynomial expressions across each other.
In our exercise, the distributive property guides us through the multiplication of two binomials, \((3x^2-5)\) and \((2x^2+1)\). Here’s how it's utilized step-by-step:

• Multiply each term of the first polynomial by each term of the second polynomial: This gives the sequence of products: \((3x^2)(2x^2)\), \((3x^2)(1)\), \((-5)(2x^2)\), \((-5)(1)\).
• Apply the multiplication and simplification: Applying the distributive property and exponent rules helps simplify these to: \(6x^4\), \(3x^2\), \(-10x^2\), \(-5\).

Finally, it is the combination of these terms following multiplication that simplifies to the final result: \(6x^4 - 7x^2 - 5\).
Understanding and using the distributive property effectively is crucial for tackling and simplifying complex polynomial equations.